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

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	*Reference is also made to the chapter  [[Theory_of_Stochastic_Signals/Gaussian_Distributed_Random_Variables\|"Gaussian distributed random variables"]]  in the book  "Stochastic Signal Theory".		*Reference is also made to the chapter  [[Theory_of_Stochastic_Signals/Gaussian_Distributed_Random_Variables\|"Gaussian distributed random variables"]]  in the book  "Stochastic Signal Theory".

	*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/Bounds_for_Block_Error_Probability#The_upper_bound_according_to_Bhattacharyya\|"Bhattacharyya Bound"]]  for the AWGN channel.

	* Further we refer to the interactive HTML5/JavaScript 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"]].
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	[[Category:Channel Coding: Exercises\|^1.6 Error Probability Bounds^]]		[[Category:Channel Coding: Exercises\|^1.6 Error Probability Bounds^]]
	[[de:Aufgaben:1.~~16Z_Schranken_für_Q(x)~~]]		[[de:Aufgaben:Aufgabe 1.16Z: Schranken für die Gaußsche Fehlerfunktion]]

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	[[Category:Channel Coding: Exercises\|^1.6 Error Probability Bounds^]]		[[Category:Channel Coding: Exercises\|^1.6 Error Probability Bounds^]]
			[[de:Aufgaben:1.16Z_Schranken_für_Q(x)]]

Hwang at 16:03, 23 January 2023

2023-01-23T16:03:56Z

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	*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  $x$  is,  the more inaccurately  ${\rm Q}(x)$  is approximated by  ${\rm Q}_{\rm CR}(x)$.

	*When looking at the graph on the information ~~page~~,  I first had the impression that  ${\rm Q}_{\rm CR}(x)$  results from  ${\rm Q}(x)$  by shifting to the right or shifting up.		*When looking at the graph in the information section,  I first had the impression that  ${\rm Q}_{\rm CR}(x)$  results from  ${\rm Q}(x)$  by shifting to the right or shifting up.

	*But this is only an optical illusion and does not correspond to the facts.		*But this is only an optical illusion and does not correspond to the facts.

Hwang at 06:19, 18 September 2022

2022-09-18T06:19:47Z

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	[[File:P_ID2415__KC_A_1_15.png\|right\|frame\|Function  ${\rm Q}(x)$  and approximations;<br>it holds:  ${\rm Q_u}(x)\le{\rm Q}(x)\le{\rm Q_o}(x)$]]		[[File:P_ID2415__KC_A_1_15.png\|right\|frame\|Function  ${\rm Q}(x)$  and approximations;<br>it holds:  ${\rm Q_u}(x)\le{\rm Q}(x)\le{\rm Q_o}(x)$]]

	The probability that a zero-mean Gaussian random variable  $n$  with standard deviation  $\sigma$   ⇒   variance  $\sigma^2$  is greater in ~~amount~~ than a given value  $A$  is equal to		The probability that a zero-mean Gaussian random variable  $n$  with standard deviation  $\sigma$   ⇒   variance  $\sigma^2$  is greater in magnitude than a given value  $A$  is equal to

	:$${\rm Pr}(n > A) = {\rm Pr}(n < -A) ={\rm Q}(A/\sigma) \hspace{0.05cm}.$$		:$${\rm Pr}(n > A) = {\rm Pr}(n < -A) ={\rm Q}(A/\sigma) \hspace{0.05cm}.$$

Guenter at 15:33, 5 August 2022

2022-08-05T15:33:47Z

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	:$${\rm Q_o}(x)=\frac{1}{\sqrt{\rm 2\pi}\cdot x}\cdot {\rm e}^{-x^{\rm 2}/\rm 2} \hspace{0.3cm} \Rightarrow \hspace{0.3cm} {\rm Q_o}(4 )=\frac{1}{\sqrt{\rm 2\pi}\cdot 4}\cdot {\rm e}^{-8 }\hspace{0.15cm}\underline{\approx 3.346 \cdot 10^{-5}}\hspace{0.05cm}.$$		:$${\rm Q_o}(x)=\frac{1}{\sqrt{\rm 2\pi}\cdot x}\cdot {\rm e}^{-x^{\rm 2}/\rm 2} \hspace{0.3cm} \Rightarrow \hspace{0.3cm} {\rm Q_o}(4 )=\frac{1}{\sqrt{\rm 2\pi}\cdot 4}\cdot {\rm e}^{-8 }\hspace{0.15cm}\underline{\approx 3.346 \cdot 10^{-5}}\hspace{0.05cm}.$$

	The lower bound can be converted as follows:		*The lower bound can be converted as follows:

	:$${\rm Q_u}( x)=(1-1/x^2) \cdot {\rm Q_o}(x) \hspace{0.3cm} \Rightarrow \hspace{0.3cm} {\rm Q_u}(4 ) \hspace{0.15cm}\underline{\approx 3.137 \cdot 10^{-5}} \hspace{0.05cm}.$$		:$${\rm Q_u}( x)=(1-1/x^2) \cdot {\rm Q_o}(x) \hspace{0.3cm} \Rightarrow \hspace{0.3cm} {\rm Q_u}(4 ) \hspace{0.15cm}\underline{\approx 3.137 \cdot 10^{-5}} \hspace{0.05cm}.$$

	*The relative deviations from the ~~"real"~~ value ${\rm Q}(4) = 3.167 · 10^{–5}$ ~~sind~~ $+5\%$ ~~bzw~~. $–1\%$.		*The relative deviations from the  actual  value  ${\rm Q}(4) = 3.167 · 10^{–5}$  are  $+5\%$  resp.  $–1\%$.



	'''(2)'''  Correct are the <u>solutions 1 and 2</u>:		'''(2)'''  Correct are the  <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  $x = 2$,  the actual function value  ${\rm Q}(x) = 2.275 \cdot 10^{-2}$  is bounded by  ${\rm Q_{o}}(x) = 2.7 \cdot 10^{-2}$  and  ${\rm Q_u}(x) = 2.025 \cdot 10^{-2}$, respectively.
	*The relative deviations are therefore $18.7\%$ ~~and~~ $-11\%,$ ~~respectively~~.
	*The last statement is wrong:   Only for $x < 0.37$ ${\rm Q_o}(x) > 1$ is valid.		*The relative deviations are therefore  $18.7\%$ resp.  $-11\%,$.

			*The last statement is wrong:   Only for  $x < 0.37$   ⇒   ${\rm Q_o}(x) > 1$  is valid.




	'''(3)'''  For the quotient of ${\rm Q}_{\rm CR}(x)$ and ${\rm Q_o}(x)$, according to the given equations:		'''(3)'''  For the quotient of  ${\rm Q}_{\rm CR}(x)$  and  ${\rm Q_o}(x)$,  according to the given equations:

	:$$q(x) = \frac{{\rm Q_{CR}}(x)}{{\rm Q_{o}}(x)} = \frac{{\rm exp}(-x^2/2)}{{\rm exp}(-x^2/2)/({\sqrt{2\pi} \cdot x})} = {\sqrt{2\pi} \cdot x}$$		:$$q(x) = \frac{{\rm Q_{CR}}(x)}{{\rm Q_{o}}(x)} = \frac{{\rm exp}(-x^2/2)}{{\rm exp}(-x^2/2)/({\sqrt{2\pi} \cdot x})} = {\sqrt{2\pi} \cdot x}$$
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	:$$\Rightarrow \hspace{0.3cm} q(x) \approx 2.5 \cdot x \hspace{0.3cm} \Rightarrow \hspace{0.3cm} q(x =2) \hspace{0.15cm}\underline{=5}\hspace{0.05cm}, \hspace{0.2cm}q(x =4)\hspace{0.15cm}\underline{=10}\hspace{0.05cm}, \hspace{0.2cm}q(x =6) \hspace{0.15cm}\underline{=15}\hspace{0.05cm}.$$		:$$\Rightarrow \hspace{0.3cm} q(x) \approx 2.5 \cdot x \hspace{0.3cm} \Rightarrow \hspace{0.3cm} q(x =2) \hspace{0.15cm}\underline{=5}\hspace{0.05cm}, \hspace{0.2cm}q(x =4)\hspace{0.15cm}\underline{=10}\hspace{0.05cm}, \hspace{0.2cm}q(x =6) \hspace{0.15cm}\underline{=15}\hspace{0.05cm}.$$

	*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  $x$  is,  the more inaccurately  ${\rm Q}(x)$  is approximated by  ${\rm Q}_{\rm CR}(x)$.
	*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.
			*When looking at the graph on the information page,  I first had the impression that  ${\rm Q}_{\rm CR}(x)$  results from  ${\rm Q}(x)$  by shifting to the right or shifting up.

			*But this is only an optical illusion and does not correspond to the facts.




	'''(4)'''  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)'''  With  $\underline{K = 0.5}$  the new bound  $0.5 \cdot {\rm Q}_{\rm CR}(x)$  for  $x = 0$  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,  the falsification  $q \approx 1.25 \cdot x$  thus also becomes only half as large.
	{{ML-Fuß}}		{{ML-Fuß}}

Guenter at 15:18, 5 August 2022

2022-08-05T15:18:08Z

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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"]].

Guenter at 15:15, 5 August 2022

2022-08-05T15:15:47Z

← Older revision		Revision as of 17:15, 5 August 2022
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	Hints:		Hints:
	* This exercise belongs to the chapter  [[Channel_Coding/Bounds_for_Block_Error_Probability\|"Bounds for ~~Block Error Probability~~"]].		* This exercise belongs to the chapter  [[Channel_Coding/Bounds_for_Block_Error_Probability\|"Bounds for block error probability"]].

	*Reference is also made to the chapter  [[Theory_of_Stochastic_Signals/Gaussian_Distributed_Random_Variables\|"Gaussian distributed random variables"]]  in the book  "Stochastic Signal Theory".		*Reference is also made to the chapter  [[Theory_of_Stochastic_Signals/Gaussian_Distributed_Random_Variables\|"Gaussian distributed random variables"]]  in the book  "Stochastic Signal Theory".

Guenter at 15:14, 5 August 2022

2022-08-05T15:14:29Z

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	{{quiz-Header\|Buchseite=Channel_Coding/Limits_for_Block_Error_Probability}}		{{quiz-Header\|Buchseite=Channel_Coding/Limits_for_Block_Error_Probability}}

	[[File:P_ID2415__KC_A_1_15.png\|right\|frame\|${\rm Q}(x)$  and ~~related functions~~]]		[[File:P_ID2415__KC_A_1_15.png\|right\|frame\|Function  ${\rm Q}(x)$  and approximations;<br>it holds:  ${\rm Q_u}(x)\le{\rm Q}(x)\le{\rm Q_o}(x)$]]

	The probability that a zero mean Gaussian random variable  $n$  with standard deviation  $\sigma$   ⇒   variance  $\sigma^2$  is greater in amount than a given value  $A$ is equal to		The probability that a zero-mean Gaussian random variable  $n$  with standard deviation  $\sigma$   ⇒   variance  $\sigma^2$  is greater in amount than a given value  $A$  is equal to

	:$${\rm Pr}(n > A) = {\rm Pr}(n < -A) ={\rm Q}(A/\sigma) \hspace{0.05cm}.$$		:$${\rm Pr}(n > A) = {\rm Pr}(n < -A) ={\rm Q}(A/\sigma) \hspace{0.05cm}.$$

	Here is used one of the most important functions for ~~communications engineering~~ (drawn in red in the diagram):  <br>the  [[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  (drawn in red in the diagram):  <br>the  [[Theory_of_Stochastic_Signals/Gaussian_Distributed_Random_Variables#Exceedance_probability\|"complementary Gaussian error function"]]

	:$${\rm Q} (x) = \frac{\rm 1}{\sqrt{\rm 2\pi}}\int_{\it x}^{+\infty}\rm e^{\it -u^{\rm 2}/\rm 2}\,d \it u \hspace{0.05cm}.$$		:$${\rm Q} (x) = \frac{\rm 1}{\sqrt{\rm 2\pi}}\int_{\it x}^{+\infty}\rm e^{\it -u^{\rm 2}/\rm 2}\,d \it u \hspace{0.05cm}.$$

	${\rm Q}(x)$ 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 $\to 0$.


	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 ${\rm Q}$–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 upper bound $($~~upper~~ blue curve in adjacent graph, ~~only~~ valid for  $x > 0)$:		*the  "upper bound"   ⇒   upper   $($German:  "obere"   ⇒   subscript: "o"$)$  blue curve in adjacent graph,  valid for  $x > 0$:

	:$$ {\rm Q_o}(x)=\frac{\rm 1}{\sqrt{\rm 2\pi}\cdot x}\cdot {\rm e}^{-x^{\rm 2}/\rm 2}\hspace{0.15cm} \ge \hspace{0.15cm} {\rm Q} (x) \hspace{0.05cm},$$		:$$ {\rm Q_o}(x)=\frac{\rm 1}{\sqrt{\rm 2\pi}\cdot x}\cdot {\rm e}^{-x^{\rm 2}/\rm 2}\hspace{0.15cm} \ge \hspace{0.15cm} {\rm Q} (x) \hspace{0.05cm},$$

	*the lower bound $($~~lower~~ blue curve in ~~the~~ graph, ~~only~~ valid for  $x > 1)$:		*the  "lower bound"   ⇒   upper   $($German:  "untere"   ⇒   subscript: "u"$)$  blue curve in adjacent graph,  valid for  $x > 1$:

	:$$ {\rm Q_u}(x)=\frac{\rm 1-{\rm 1}/{\it x^{\rm 2}}}{\sqrt{\rm 2\pi}\cdot x}\cdot \rm e^{-x^{\rm 2}/\rm 2} \hspace{0.15cm} \le \hspace{0.15cm} {\rm Q} (x) \hspace{0.05cm},$$		:$$ {\rm Q_u}(x)=\frac{\rm 1-{\rm 1}/{\it x^{\rm 2}}}{\sqrt{\rm 2\pi}\cdot x}\cdot \rm e^{-x^{\rm 2}/\rm 2} \hspace{0.15cm} \le \hspace{0.15cm} {\rm Q} (x) \hspace{0.05cm},$$

	*the Chernoff-Rubin bound $($green curve in the graph, drawn for  $K = 1)$:		*the  "Chernoff-Rubin bound"  $($green curve in the graph, drawn for  $K = 1)$:

	:$${\rm Q_{CR}}(x)=K \cdot {\rm e}^{-x^{\rm 2}/\rm 2} \hspace{0.15cm} \ge \hspace{0.15cm} {\rm Q} (x) \hspace{0.05cm}.$$		:$${\rm Q_{CR}}(x)=K \cdot {\rm e}^{-x^{\rm 2}/\rm 2} \hspace{0.15cm} \ge \hspace{0.15cm} {\rm Q} (x) \hspace{0.05cm}.$$
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			Hints:
			* This exercise belongs to the chapter  [[Channel_Coding/Bounds_for_Block_Error_Probability\|"Bounds for Block Error Probability"]].

			*Reference is also made to the chapter  [[Theory_of_Stochastic_Signals/Gaussian_Distributed_Random_Variables\|"Gaussian distributed random variables"]]  in the book  "Stochastic Signal Theory".

			*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"]].


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


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	<quiz display=simple>		<quiz display=simple>

	{What values do the upper and lower bounds for  $x = 4$ provide?		{What values do the upper and lower bounds for  $x = 4$  provide?
	\|type="{}"}		\|type="{}"}
	${\rm Q_{o}}(x = 4) \ = \ $ { 3.346 3% }$\ \cdot 10^{-5} $		${\rm Q_{o}}(x = 4) \ = \ $ { 3.346 3% }$\ \cdot 10^{-5} $
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	{What statements hold for the functions  ${\rm Q_{o}}(x)$  and  ${\rm Q_{u}}(x)$?		{What statements hold for the functions  ${\rm Q_{o}}(x)$  and  ${\rm Q_{u}}(x)$?
	\|type="[]"}		\|type="[]"}
	+ For  $x ≥ 2$  ~~the two~~ bounds are usable.		+ For  $x ≥ 2$:  Both bounds are usable.
	+ For  $x < 1$  ~~is ~~ ${\rm Q_{u}}(x)$  unusable  $($because  ${\rm Q_{u}}(x)< 0)$.		+ For  $x < 1$:  ${\rm Q_{u}}(x)$  is unusable   $($because  ${\rm Q_{u}}(x)< 0)$.
	- For  $x < 1$  ~~is ~~ ${\rm Q_{o}}(x)$  unusable  $($because  ${\rm Q_{o}}(x)> 1)$.		- For  $x < 1$:  ${\rm Q_{o}}(x)$  is unusable  $($because  ${\rm Q_{o}}(x)> 1)$.


	{By what factor is the Chernoff-Rubin ~~bound~~ above  ${\rm Q_{o}}(x)$?		{By what factor is the Chernoff-Rubin Bound above  ${\rm Q_{o}}(x)$?
	\|type="{}"}		\|type="{}"}
	${\rm Q}_{\rm CR}(x = 2)/{\rm Q_{o}}(x = 2 ) \ = \ $ { 5 3% }		${\rm Q}_{\rm CR}(x = 2)/{\rm Q_{o}}(x = 2 ) \ = \ $ { 5 3% }
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	${\rm Q}_{\rm CR}(x = 6)/{\rm Q_{o}}(x = 6 ) \ = \ $ { 15 3% }		${\rm Q}_{\rm CR}(x = 6)/{\rm Q_{o}}(x = 6 ) \ = \ $ { 15 3% }

	{Determine  $K$  such that  $K \cdot {\rm Q}_{\rm CR}(x)$  as close as possible to  ${\rm Q}(x)$  and at the same time  ${\rm Q}(x) ≤ K · {\rm Q}_{\rm CR}(x)$  is observed for all  $x > 0$ .		{Determine  $K$  such that  $K \cdot {\rm Q}_{\rm CR}(x)$  is as close as possible to  ${\rm Q}(x)$  and at the same time  ${\rm Q}(x) ≤ K · {\rm Q}_{\rm CR}(x)$  is observed for all  $x > 0$ .
	\|type="{}"}		\|type="{}"}
	$K \ = \ $ { 0.5 3% }		$K \ = \ $ { 0.5 3% }

Noah at 18:24, 30 July 2022

2022-07-30T18:24:44Z

← Older revision		Revision as of 20:24, 30 July 2022
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	Hints:		Hints:
	*This exercise belongs to the chapter  [[Channel_Coding/Limits_for_Block_Error_Probability\|block error probability bounds]].		*This exercise belongs to the chapter  [[Channel_Coding/Limits_for_Block_Error_Probability\|"block error probability bounds"]].
	*Reference is also made to the chapter  [[Theory_of_Stochastic_Signals/Gaussian_Distributed_Random_Variables\|Gaussian distributed random variables]]  in the book "Stochastic Signal Theory".		*Reference is also made to the chapter  [[Theory_of_Stochastic_Signals/Gaussian_Distributed_Random_Variables\|"Gaussian distributed random variables"]]  in the book "Stochastic Signal Theory".
	*The exercise also provides some important hints for solving  [[Aufgaben:Exercise_1.16:_Block_Error_Probability_Bounds_for_AWGN\|Exercise 1.16]], in which the function  ${\rm Q}_{\rm CR}(x)$  is used to derive  [[Channel_Coding/Limits_for_Block_Error_Probability#The_upper_bound_according_to_Bhattacharyya\|Bhattacharyya barrier]]  is required for the AWGN channel.		*The exercise also provides some important hints for solving  [[Aufgaben:Exercise_1.16:_Block_Error_Probability_Bounds_for_AWGN\|"Exercise 1.16"]], in which the function  ${\rm Q}_{\rm CR}(x)$  is used to derive  [[Channel_Coding/Limits_for_Block_Error_Probability#The_upper_bound_according_to_Bhattacharyya\|"Bhattacharyya barrier"]]  is required for the AWGN channel.
	*Further we refer to the interactive applet  [[Applets:Complementary_Gaussian_Error_Functions\|Complementary Gaussian Error Functions]].		*Further we refer to the interactive applet  [[Applets:Complementary_Gaussian_Error_Functions\|"Complementary Gaussian Error Functions"]].

Noah at 18:23, 30 July 2022

2022-07-30T18:23:47Z

← Older revision		Revision as of 20:23, 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 $\to 0$.

← Older revision		Revision as of 20:24, 30 July 2022
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	Hints:		Hints:
	*This exercise belongs to the chapter  [[Channel_Coding/Limits_for_Block_Error_Probability\|block error probability bounds]].		*This exercise belongs to the chapter  [[Channel_Coding/Limits_for_Block_Error_Probability\|"block error probability bounds"]].
	*Reference is also made to the chapter  [[Theory_of_Stochastic_Signals/Gaussian_Distributed_Random_Variables\|Gaussian distributed random variables]]  in the book "Stochastic Signal Theory".		*Reference is also made to the chapter  [[Theory_of_Stochastic_Signals/Gaussian_Distributed_Random_Variables\|"Gaussian distributed random variables"]]  in the book "Stochastic Signal Theory".
	*The exercise also provides some important hints for solving  [[Aufgaben:Exercise_1.16:_Block_Error_Probability_Bounds_for_AWGN\|Exercise 1.16]], in which the function  ${\rm Q}_{\rm CR}(x)$  is used to derive  [[Channel_Coding/Limits_for_Block_Error_Probability#The_upper_bound_according_to_Bhattacharyya\|Bhattacharyya barrier]]  is required for the AWGN channel.		*The exercise also provides some important hints for solving  [[Aufgaben:Exercise_1.16:_Block_Error_Probability_Bounds_for_AWGN\|"Exercise 1.16"]], in which the function  ${\rm Q}_{\rm CR}(x)$  is used to derive  [[Channel_Coding/Limits_for_Block_Error_Probability#The_upper_bound_according_to_Bhattacharyya\|"Bhattacharyya barrier"]]  is required for the AWGN channel.
	*Further we refer to the interactive applet  [[Applets:Complementary_Gaussian_Error_Functions\|Complementary Gaussian Error Functions]].		*Further we refer to the interactive applet  [[Applets:Complementary_Gaussian_Error_Functions\|"Complementary Gaussian Error Functions"]].