Aufgaben:Exercise 1.16Z: Bounds for the Gaussian Error Function - Revision history

Hwang at 16:03, 23 January 2023

2023-01-23T16:03:56Z

Hwang at 06:19, 18 September 2022

2022-09-18T06:19:47Z

Guenter at 15:33, 5 August 2022

2022-08-05T15:33:47Z

Guenter at 15:18, 5 August 2022

2022-08-05T15:18:08Z

Guenter at 15:15, 5 August 2022

2022-08-05T15:15:47Z

Guenter at 15:14, 5 August 2022

2022-08-05T15:14:29Z

Noah at 18:24, 30 July 2022

2022-07-30T18:24:44Z

Noah at 18:23, 30 July 2022

2022-07-30T18:23:47Z

Noah at 18:23, 30 July 2022

2022-07-30T18:23:22Z

Noah at 17:51, 30 July 2022

2022-07-30T17:51:06Z

@@ Line 105: / Line 105: @@
 *The larger the abscissa value&nbsp; $x$&nbsp; is,&nbsp; the more inaccurately&nbsp; ${\rm Q}(x)$&nbsp; is approximated by&nbsp; ${\rm Q}_{\rm CR}(x)$.
-*When looking at the graph on the information page,&nbsp; I first had the impression that&nbsp; ${\rm Q}_{\rm CR}(x)$&nbsp; results from&nbsp; ${\rm Q}(x)$&nbsp; by shifting to the right or shifting up.&nbsp;
+*When looking at the graph in the information section,&nbsp; I first had the impression that&nbsp; ${\rm Q}_{\rm CR}(x)$&nbsp; results from&nbsp; ${\rm Q}(x)$&nbsp; by shifting to the right or shifting up.&nbsp;
 *But this is only an optical illusion and does not correspond to the facts.

@@ Line 3: / Line 3: @@
 [[File:P_ID2415__KC_A_1_15.png|right|frame|Function&nbsp; ${\rm Q}(x)$&nbsp; and approximations;<br>it holds:&nbsp; ${\rm Q_u}(x)\le{\rm Q}(x)\le{\rm Q_o}(x)$]]
-The probability that a zero-mean Gaussian random variable&nbsp; $n$&nbsp; with standard deviation&nbsp; $\sigma$ &nbsp; &rArr; &nbsp; variance&nbsp; $\sigma^2$&nbsp; is greater in amount than a given value&nbsp; $A$&nbsp; is equal to
+The probability that a zero-mean Gaussian random variable&nbsp; $n$&nbsp; with standard deviation&nbsp; $\sigma$ &nbsp; &rArr; &nbsp; variance&nbsp; $\sigma^2$&nbsp; is greater in magnitude than a given value&nbsp; $A$&nbsp; is equal to
 :$${\rm Pr}(n > A) = {\rm Pr}(n < -A) ={\rm Q}(A/\sigma) \hspace{0.05cm}.$$

@@ Line 79: / Line 79: @@
 :$${\rm Q_o}(x)=\frac{1}{\sqrt{\rm 2\pi}\cdot x}\cdot {\rm e}^{-x^{\rm 2}/\rm 2} \hspace{0.3cm} \Rightarrow \hspace{0.3cm} {\rm Q_o}(4 )=\frac{1}{\sqrt{\rm 2\pi}\cdot 4}\cdot {\rm e}^{-8 }\hspace{0.15cm}\underline{\approx 3.346 \cdot 10^{-5}}\hspace{0.05cm}.$$
-The lower bound can be converted as follows:
+*The lower bound can be converted as follows:
 :$${\rm Q_u}( x)=(1-1/x^2) \cdot {\rm Q_o}(x) \hspace{0.3cm} \Rightarrow \hspace{0.3cm} {\rm Q_u}(4 ) \hspace{0.15cm}\underline{\approx 3.137 \cdot 10^{-5}} \hspace{0.05cm}.$$
-*The relative deviations from the "real" value ${\rm Q}(4) = 3.167 · 10^{–5}$ sind $+5\%$ bzw. $–1\%$.
+*The relative deviations from the&nbsp; actual&nbsp; value&nbsp; ${\rm Q}(4) = 3.167 · 10^{–5}$&nbsp; are&nbsp; $+5\%$&nbsp; resp.&nbsp; $–1\%$.
-'''(2)'''&nbsp; Correct are the <u>solutions 1 and 2</u>:
+'''(2)'''&nbsp; Correct are the&nbsp; <u>solutions 1 and 2</u>:
-*For $x = 2$, the actual function value ${\rm Q}(x) = 2.275 - 10^{-2}$ is bounded by ${\rm Q_{o}}(x) = 2.7 - 10^{-2}$ and ${\rm Q_u}(x) = 2.025 - 10^{-2}$, respectively.
+*For&nbsp; $x = 2$,&nbsp; the actual function value&nbsp; ${\rm Q}(x) = 2.275 \cdot 10^{-2}$&nbsp; is bounded by&nbsp; ${\rm Q_{o}}(x) = 2.7 \cdot 10^{-2}$&nbsp; and&nbsp; ${\rm Q_u}(x) = 2.025 \cdot 10^{-2}$, respectively.
-*The relative deviations are therefore $18.7\%$ and $-11\%,$ respectively.
-*The last statement is wrong: &nbsp; Only for $x < 0.37$ ${\rm Q_o}(x) > 1$ is valid.
+*The relative deviations are therefore&nbsp; $18.7\%$&nbsp;resp.&nbsp; $-11\%,$.
-'''(3)'''&nbsp; For the quotient of ${\rm Q}_{\rm CR}(x)$ and ${\rm Q_o}(x)$, according to the given equations:
+'''(3)'''&nbsp; For the quotient of&nbsp; ${\rm Q}_{\rm CR}(x)$&nbsp; and&nbsp; ${\rm Q_o}(x)$,&nbsp; according to the given equations:
 :$$q(x) = \frac{{\rm Q_{CR}}(x)}{{\rm Q_{o}}(x)} = \frac{{\rm exp}(-x^2/2)}{{\rm exp}(-x^2/2)/({\sqrt{2\pi} \cdot x})} = {\sqrt{2\pi} \cdot x}$$
@@ Line 101: / Line 103: @@
 :$$\Rightarrow \hspace{0.3cm} q(x) \approx 2.5 \cdot x \hspace{0.3cm} \Rightarrow \hspace{0.3cm} q(x =2) \hspace{0.15cm}\underline{=5}\hspace{0.05cm}, \hspace{0.2cm}q(x =4)\hspace{0.15cm}\underline{=10}\hspace{0.05cm}, \hspace{0.2cm}q(x =6) \hspace{0.15cm}\underline{=15}\hspace{0.05cm}.$$
-*The larger the abscissa value $x$ is, the more inaccurately ${\rm Q}(x)$ is approximated by ${\rm Q}_{\rm CR}(x)$.
+*The larger the abscissa value&nbsp; $x$&nbsp; is,&nbsp; the more inaccurately&nbsp; ${\rm Q}(x)$&nbsp; is approximated by&nbsp; ${\rm Q}_{\rm CR}(x)$.
-*When looking at the graph on the information page, one has (I had) the impression that ${\rm Q}_{\rm CR}(x)$ results from ${\rm Q}(x)$ by shifting down or shifting up. But this is only an optical illusion and does not correspond to the facts.
-'''(4)'''&nbsp; With $\underline{K = 0.5}$ the new bound $0.5 - {\rm Q}_{\rm CR}(x)$ for $x = 0$ agrees exactly with ${\rm Q}(x=0) = 0.500$.
+'''(4)'''&nbsp; With&nbsp; $\underline{K = 0.5}$&nbsp; the new bound&nbsp; $0.5 \cdot {\rm Q}_{\rm CR}(x)$&nbsp; for&nbsp; $x = 0$&nbsp; agrees exactly with ${\rm Q}(x=0) = 0.500$.
-*For larger abscissa values, the corruption $q \approx 1.25 - x$ thus also becomes only half as large.
+*For larger abscissa values,&nbsp; the falsification&nbsp; $q \approx 1.25 \cdot x$&nbsp; thus also becomes only half as large.
 {{ML-Fuß}}

← Older revision		Revision as of 15:18, 5 August 2022
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	*The exercise provides some important hints for solving  [[Aufgaben:Exercise_1.16:_Block_Error_Probability_Bounds_for_AWGN\|"Exercise 1.16"]],  in which  ${\rm Q}_{\rm CR}(x)$  is used to derive the   [[Channel_Coding/Limits_for_Block_Error_Probability#The_upper_bound_according_to_Bhattacharyya\|"Bhattacharyya Bound"]]  for the AWGN channel.		*The exercise provides some important hints for solving  [[Aufgaben:Exercise_1.16:_Block_Error_Probability_Bounds_for_AWGN\|"Exercise 1.16"]],  in which  ${\rm Q}_{\rm CR}(x)$  is used to derive the   [[Channel_Coding/Limits_for_Block_Error_Probability#The_upper_bound_according_to_Bhattacharyya\|"Bhattacharyya Bound"]]  for the AWGN channel.

−	* Further we refer to the interactive applet  [[Applets:Komplementäre_Gaußsche_Fehlerfunktionen\| "Complementary Gaussian error functions"]].	+	* Further we refer to the interactive HTML5/JavaScript applet  [[Applets:Komplementäre_Gaußsche_Fehlerfunktionen\| "Complementary Gaussian error functions"]].

@@ Line 33: / Line 33: @@
 Hints:
-* This exercise belongs to the chapter&nbsp; [[Channel_Coding/Bounds_for_Block_Error_Probability|"Bounds for Block Error Probability"]].
+* This exercise belongs to the chapter&nbsp; [[Channel_Coding/Bounds_for_Block_Error_Probability|"Bounds for block error probability"]].
 *Reference is also made to the chapter&nbsp; [[Theory_of_Stochastic_Signals/Gaussian_Distributed_Random_Variables|"Gaussian distributed random variables"]]&nbsp; in the book&nbsp; "Stochastic Signal Theory".

@@ Line 1: / Line 1: @@
 {{quiz-Header|Buchseite=Channel_Coding/Limits_for_Block_Error_Probability}}
-[[File:P_ID2415__KC_A_1_15.png|right|frame|${\rm Q}(x)$&nbsp; and related functions]]
+[[File:P_ID2415__KC_A_1_15.png|right|frame|Function&nbsp; ${\rm Q}(x)$&nbsp; and approximations;<br>it holds:&nbsp; ${\rm Q_u}(x)\le{\rm Q}(x)\le{\rm Q_o}(x)$]]
-The probability that a zero mean Gaussian random variable&nbsp; $n$&nbsp; with standard deviation&nbsp; $\sigma$ &nbsp; &rArr; &nbsp; variance&nbsp; $\sigma^2$&nbsp; is greater in amount than a given value&nbsp; $A$ is equal to
+The probability that a zero-mean Gaussian random variable&nbsp; $n$&nbsp; with standard deviation&nbsp; $\sigma$ &nbsp; &rArr; &nbsp; variance&nbsp; $\sigma^2$&nbsp; is greater in amount than a given value&nbsp; $A$&nbsp; is equal to
 :$${\rm Pr}(n > A) = {\rm Pr}(n < -A) ={\rm Q}(A/\sigma) \hspace{0.05cm}.$$
-Here is used one of the most important functions for communications engineering (drawn in red in the diagram): &nbsp;<br>the&nbsp; [[Theory_of_Stochastic_Signals/Gaussian_Distributed_Random_Variables#Exceedance_probability|"complementary Gaussian error function"]]
+Here is used one of the most important functions for Communications Engineering&nbsp; (drawn in red in the diagram): &nbsp;<br>the&nbsp; [[Theory_of_Stochastic_Signals/Gaussian_Distributed_Random_Variables#Exceedance_probability|"complementary Gaussian error function"]]
 :$${\rm Q} (x) = \frac{\rm 1}{\sqrt{\rm 2\pi}}\int_{\it x}^{+\infty}\rm e^{\it -u^{\rm 2}/\rm 2}\,d \it u \hspace{0.05cm}.$$
-${\rm Q}(x)$ is a monotonically decreasing function with ${\rm Q}(0) = 0.5$. For very large values of $x$, ${\rm Q}(x)$ tends $\to 0$.
+${\rm Q}(x)$&nbsp; is a monotonically decreasing function with&nbsp; ${\rm Q}(0) = 0.5$.&nbsp; For very large values of&nbsp; $x$ &nbsp; &rArr; &nbsp;  ${\rm Q}(x)$ tends $\to 0$.
-The integral of the ${\rm Q}$ function is not analytically solvable and is usually given in tabular form. From the literature, however, manageable approximations or bounds for positive $x$ values are known:
+The integral of the ${\rm Q}$&ndash;function is not analytically solvable and is usually given in tabular form.&nbsp; From the literature,&nbsp; however,&nbsp; manageable approximations or bounds for positive&nbsp; $x$&nbsp; values are known:
-*the upper bound $($upper blue curve in adjacent graph, only valid for&nbsp; $x > 0)$:
+*the&nbsp;  "upper bound" &nbsp; &rArr; &nbsp; upper &nbsp; $($German:&nbsp; "obere" &nbsp; &rArr; &nbsp; subscript: "o"$)$&nbsp; blue curve in adjacent graph,&nbsp; valid for&nbsp; $x > 0$:
 :$$ {\rm Q_o}(x)=\frac{\rm 1}{\sqrt{\rm 2\pi}\cdot x}\cdot {\rm e}^{-x^{\rm 2}/\rm 2}\hspace{0.15cm} \ge \hspace{0.15cm} {\rm Q} (x) \hspace{0.05cm},$$
-*the lower bound $($lower blue curve in the graph, only valid for&nbsp; $x > 1)$:
+*the&nbsp;  "lower bound" &nbsp; &rArr; &nbsp; upper &nbsp; $($German:&nbsp; "untere" &nbsp; &rArr; &nbsp; subscript: "u"$)$&nbsp; blue curve in adjacent graph,&nbsp; valid for&nbsp; $x > 1$:
 :$$ {\rm Q_u}(x)=\frac{\rm 1-{\rm 1}/{\it x^{\rm 2}}}{\sqrt{\rm 2\pi}\cdot x}\cdot \rm e^{-x^{\rm 2}/\rm 2} \hspace{0.15cm} \le \hspace{0.15cm}  {\rm Q} (x) \hspace{0.05cm},$$
-*the Chernoff-Rubin bound $($green curve in the graph, drawn for&nbsp; $K = 1)$:
+*the&nbsp; "Chernoff-Rubin bound"&nbsp; $($green curve in the graph, drawn for&nbsp; $K = 1)$:
 :$${\rm Q_{CR}}(x)=K \cdot {\rm e}^{-x^{\rm 2}/\rm 2} \hspace{0.15cm} \ge \hspace{0.15cm} {\rm Q} (x) \hspace{0.05cm}.$$
@@ Line 33: / Line 32: @@
@@ Line 50: / Line 50: @@
 <quiz display=simple>
-{What values do the upper and lower bounds for&nbsp; $x = 4$ provide?
+{What values do the upper and lower bounds for&nbsp; $x = 4$&nbsp; provide?
 |type="{}"}
 ${\rm Q_{o}}(x = 4) \ = \ $ { 3.346 3% }$\ \cdot 10^{-5} $
@@ Line 57: / Line 57: @@
 {What statements hold for the functions&nbsp; ${\rm Q_{o}}(x)$&nbsp; and&nbsp; ${\rm Q_{u}}(x)$?
 |type="[]"}
-+ For&nbsp; $x ≥ 2$&nbsp; the two bounds are usable.
++ For&nbsp; $x ≥ 2$:&nbsp; Both bounds are usable.
-+ For&nbsp; $x < 1$&nbsp; is&nbsp; ${\rm Q_{u}}(x)$&nbsp; unusable&nbsp; $($because&nbsp; ${\rm Q_{u}}(x)< 0)$.
++ For&nbsp; $x < 1$:&nbsp;  ${\rm Q_{u}}(x)$&nbsp; is unusable &nbsp; $($because&nbsp; ${\rm Q_{u}}(x)< 0)$.
-- For&nbsp; $x < 1$&nbsp; is&nbsp; ${\rm Q_{o}}(x)$&nbsp; unusable&nbsp; $($because&nbsp; ${\rm Q_{o}}(x)> 1)$.
+- For&nbsp; $x < 1$:&nbsp;  ${\rm Q_{o}}(x)$&nbsp; is unusable&nbsp; $($because&nbsp; ${\rm Q_{o}}(x)> 1)$.
-{By what factor is the Chernoff-Rubin bound above&nbsp; ${\rm Q_{o}}(x)$?
+{By what factor is the Chernoff-Rubin Bound above&nbsp; ${\rm Q_{o}}(x)$?
 |type="{}"}
 ${\rm Q}_{\rm CR}(x = 2)/{\rm Q_{o}}(x = 2 )  \ = \ $ { 5 3% }
@@ Line 68: / Line 68: @@
 ${\rm Q}_{\rm CR}(x = 6)/{\rm Q_{o}}(x = 6 )  \ = \  $ { 15 3% }
-{Determine&nbsp; $K$&nbsp; such that&nbsp; $K \cdot {\rm Q}_{\rm CR}(x)$&nbsp; as close as possible to&nbsp; ${\rm Q}(x)$&nbsp; and at the same time&nbsp; ${\rm Q}(x) ≤ K · {\rm Q}_{\rm CR}(x)$&nbsp; is observed for all &nbsp;$x > 0$&nbsp;.
+{Determine&nbsp; $K$&nbsp; such that&nbsp; $K \cdot {\rm Q}_{\rm CR}(x)$&nbsp; is as close as possible to&nbsp; ${\rm Q}(x)$&nbsp; and at the same time&nbsp; ${\rm Q}(x) ≤ K · {\rm Q}_{\rm CR}(x)$&nbsp; is observed for all &nbsp;$x > 0$&nbsp;.
 |type="{}"}
 $K \ = \ $ { 0.5 3% }

@@ Line 37: / Line 37: @@
 Hints:
-*This exercise belongs to the chapter&nbsp; [[Channel_Coding/Limits_for_Block_Error_Probability|block error probability bounds]].
+*This exercise belongs to the chapter&nbsp; [[Channel_Coding/Limits_for_Block_Error_Probability|"block error probability bounds"]].
-*Reference is also made to the chapter&nbsp; [[Theory_of_Stochastic_Signals/Gaussian_Distributed_Random_Variables|Gaussian distributed random variables]]&nbsp; in the book "Stochastic Signal Theory".
+*Reference is also made to the chapter&nbsp; [[Theory_of_Stochastic_Signals/Gaussian_Distributed_Random_Variables|"Gaussian distributed random variables"]]&nbsp; in the book "Stochastic Signal Theory".
-*The exercise also provides some important hints for solving&nbsp; [[Aufgaben:Exercise_1.16:_Block_Error_Probability_Bounds_for_AWGN|Exercise 1.16]], in which the function&nbsp; ${\rm Q}_{\rm CR}(x)$&nbsp; is used to derive&nbsp; [[Channel_Coding/Limits_for_Block_Error_Probability#The_upper_bound_according_to_Bhattacharyya|Bhattacharyya barrier]]&nbsp; is required for the AWGN channel.
+*The exercise also provides some important hints for solving&nbsp; [[Aufgaben:Exercise_1.16:_Block_Error_Probability_Bounds_for_AWGN|"Exercise 1.16"]], in which the function&nbsp; ${\rm Q}_{\rm CR}(x)$&nbsp; is used to derive&nbsp; [[Channel_Coding/Limits_for_Block_Error_Probability#The_upper_bound_according_to_Bhattacharyya|"Bhattacharyya barrier"]]&nbsp; is required for the AWGN channel.
-*Further we refer to the interactive applet&nbsp; [[Applets:Complementary_Gaussian_Error_Functions|Complementary Gaussian Error Functions]].
+*Further we refer to the interactive applet&nbsp; [[Applets:Complementary_Gaussian_Error_Functions|"Complementary Gaussian Error Functions"]].

← Older revision		Revision as of 18:23, 30 July 2022
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	:$${\rm Q} (x) = \frac{\rm 1}{\sqrt{\rm 2\pi}}\int_{\it x}^{+\infty}\rm e^{\it -u^{\rm 2}/\rm 2}\,d \it u \hspace{0.05cm}.$$		:$${\rm Q} (x) = \frac{\rm 1}{\sqrt{\rm 2\pi}}\int_{\it x}^{+\infty}\rm e^{\it -u^{\rm 2}/\rm 2}\,d \it u \hspace{0.05cm}.$$

−	${\rm Q}(x)$ is a monotonically decreasing function with ${\rm Q}(0) = 0.5$. For very large values of $x$, ${\rm Q}(x) tends \to 0$.	+	${\rm Q}(x)$ is a monotonically decreasing function with ${\rm Q}(0) = 0.5$. For very large values of $x$, ${\rm Q}(x)$ tends $\to 0$.

@@ Line 1: / Line 1: @@
 {{quiz-Header|Buchseite=Channel_Coding/Limits_for_Block_Error_Probability}}
-[[File:P_ID2415__KC_A_1_15.png|right|frame|${\rm Q}(x)$&nbsp; und verwandte Funktionen]]
+[[File:P_ID2415__KC_A_1_15.png|right|frame|${\rm Q}(x)$&nbsp; and related functions]]
-Die Wahrscheinlichkeit, dass eine mittelwertfreie Gaußsche Zufallsgröße&nbsp; $n$&nbsp; mit Streuung&nbsp; $\sigma$ &nbsp; &rArr; &nbsp;  Varianz&nbsp; $\sigma^2$&nbsp; betragsmäßig größer ist als ein vorgegebener Wert&nbsp; $A$, ist gleich
+The probability that a zero mean Gaussian random variable&nbsp; $n$&nbsp; with standard deviation&nbsp; $\sigma$ &nbsp; &rArr; &nbsp; variance&nbsp; $\sigma^2$&nbsp; is greater in amount than a given value&nbsp; $A$ is equal to
 :$${\rm Pr}(n > A) = {\rm Pr}(n < -A) ={\rm Q}(A/\sigma) \hspace{0.05cm}.$$
-Hierbei verwendet ist eine der wichtigsten Funktionen für die Nachrichtentechnik (in der Grafik rot eingezeichnet): &nbsp;<br>die&nbsp; [[Theory_of_Stochastic_Signals/Gaußverteilte_Zufallsgrößen#.C3.9Cberschreitungswahrscheinlichkeit|Komplementäre Gaußsche Fehlerfunktion]]
+Here is used one of the most important functions for communications engineering (drawn in red in the diagram): &nbsp;<br>the&nbsp; [[Theory_of_Stochastic_Signals/Gaussian_Distributed_Random_Variables#Exceedance_probability|"complementary Gaussian error function"]]
 :$${\rm Q} (x) = \frac{\rm 1}{\sqrt{\rm 2\pi}}\int_{\it x}^{+\infty}\rm e^{\it -u^{\rm 2}/\rm 2}\,d \it u \hspace{0.05cm}.$$
-${\rm Q}(x)$ ist eine monoton fallende Funktion mit ${\rm Q}(0) = 0.5$. Für sehr große Werte von $x$ tendiert ${\rm Q}(x) \to 0$.
+${\rm Q}(x)$ is a monotonically decreasing function with ${\rm Q}(0) = 0.5$. For very large values of $x$, ${\rm Q}(x) tends \to 0$.
-Das Integral der&nbsp; ${\rm Q}$–Funktion ist analytisch nicht lösbar und wird meist in Tabellenform angegeben. Aus der Literatur bekannt sind aber handhabbare Näherungen bzw. Schranken für positive&nbsp; $x$–Werte:
+The integral of the ${\rm Q}$ function is not analytically solvable and is usually given in tabular form. From the literature, however, manageable approximations or bounds for positive $x$ values are known:
-*die obere Schranke $($obere blaue Kurve in nebenstehender Grafik, nur gültig für&nbsp; $x > 0)$:
+*the upper bound $($upper blue curve in adjacent graph, only valid for&nbsp; $x > 0)$:
 :$$ {\rm Q_o}(x)=\frac{\rm 1}{\sqrt{\rm 2\pi}\cdot x}\cdot {\rm e}^{-x^{\rm 2}/\rm 2}\hspace{0.15cm} \ge \hspace{0.15cm} {\rm Q} (x) \hspace{0.05cm},$$
-*die untere Schranke $($untere blaue Kurve in der Grafik, nur gültig für&nbsp; $x > 1)$:
+*the lower bound $($lower blue curve in the graph, only valid for&nbsp; $x > 1)$:
 :$$ {\rm Q_u}(x)=\frac{\rm 1-{\rm 1}/{\it x^{\rm 2}}}{\sqrt{\rm 2\pi}\cdot x}\cdot \rm e^{-x^{\rm 2}/\rm 2} \hspace{0.15cm} \le \hspace{0.15cm}  {\rm Q} (x) \hspace{0.05cm},$$
-*die Chernoff–Rubin–Schranke $($grüne Kurve in der Grafik, gezeichnet für&nbsp; $K = 1)$:
+*the Chernoff-Rubin bound $($green curve in the graph, drawn for&nbsp; $K = 1)$:
 :$${\rm Q_{CR}}(x)=K \cdot {\rm e}^{-x^{\rm 2}/\rm 2} \hspace{0.15cm} \ge \hspace{0.15cm} {\rm Q} (x) \hspace{0.05cm}.$$
-In der Aufgabe ist zu untersuchen, in wie weit diese Schranken als Näherungen für&nbsp; ${\rm Q}(x)$&nbsp; herangezogen werden können und welche Verfälschungen sich dadurch ergeben.
+In the exercise it is to be investigated to what extent these bounds can be used as approximations for&nbsp; ${\rm Q}(x)$&nbsp; and what corruptions result.
@@ Line 36: / Line 36: @@
-''Hinweise:''
+Hints:
-* Die Aufgabe bezieht sich auf das Kapitel&nbsp; [[Channel_Coding/Schranken_für_die_Blockfehlerwahrscheinlichkeit|Schranken für die Blockfehlerwahrscheinlichkeit]].
+*This exercise belongs to the chapter&nbsp; [[Channel_Coding/Limits_for_Block_Error_Probability|block error probability bounds]].
-*Bezug genommen wird auch auf das Kapitel&nbsp; [[Theory_of_Stochastic_Signals/Gaußverteilte_Zufallsgrößen|Gaußverteilte Zufallsgrößen]]&nbsp; im Buch „Stochastische Signaltheorie”.
+*Reference is also made to the chapter&nbsp; [[Theory_of_Stochastic_Signals/Gaussian_Distributed_Random_Variables|Gaussian distributed random variables]]&nbsp; in the book "Stochastic Signal Theory".
-* Die Aufgabe bietet auch einige wichtige Hinweise zur Lösung der&nbsp; [[Aufgaben:Aufgabe_1.16:_Fehlerwahrscheinlichkeitsschranken_für_AWGN|Aufgabe 1.16]], in der die Funktion&nbsp; ${\rm Q}_{\rm CR}(x)$&nbsp; zur Herleitung der&nbsp; [[Channel_Coding/Schranken_für_die_Blockfehlerwahrscheinlichkeit#Die_obere_Schranke_nach_Bhattacharyya|Bhattacharyya–Schranke]]&nbsp; für den AWGN–Kanal benötigt wird.
+*The exercise also provides some important hints for solving&nbsp; [[Aufgaben:Exercise_1.16:_Block_Error_Probability_Bounds_for_AWGN|Exercise 1.16]], in which the function&nbsp; ${\rm Q}_{\rm CR}(x)$&nbsp; is used to derive&nbsp; [[Channel_Coding/Limits_for_Block_Error_Probability#The_upper_bound_according_to_Bhattacharyya|Bhattacharyya barrier]]&nbsp; is required for the AWGN channel.
-*Weiter verweisen wir auf das interaktive Applet&nbsp; [[Applets:Komplementäre_Gaußsche_Fehlerfunktionen|Komplementäre Gaußsche Fehlerfunktionen]].
+*Further we refer to the interactive applet&nbsp; [[Applets:Complementary_Gaussian_Error_Functions|Complementary Gaussian Error Functions]].
@@ Line 46: / Line 46: @@
-===Fragebogen===
+===Questions===
 <quiz display=simple>