Aufgaben:Exercise 1.3Z: Rayleigh Fading Revisited - Revision history

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	[[Category:Mobile Communications: Exercises\|^1.2 PDF of Rayleigh Fading^]]		[[Category:Mobile Communications: Exercises\|^1.2 PDF of Rayleigh Fading^]]
	[[de:~~Exercises~~:~~Exercise_1~~.3Z:~~_Rayleigh_Fading_Revisited~~]]		[[de:Aufgaben:Exercise 1.3Z: Rayleigh Fading Revisited]]

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	* The channel  $\rm R$  $($the designation results from the color "Red" of the point cloud$)$  is Rayleigh distributed with  $\sigma_{\rm R} = 0.5$.		* The channel  $\rm R$  $($the designation results from the color "Red" of the point cloud$)$  is Rayleigh distributed with  $\sigma_{\rm R} = 0.5$.
	* The probability density functions&nbsp $\rm (PDF)$  of the magnitude  $a(t) = \|z(t)\|$  or   the squared magnitude $p(t) = \|z(t)\|^2$  are $($with  $\sigma = \sigma_{\rm R})$:		* The probability density functions&nbsp $\rm (PDF)$  of the magnitude  $a(t) = \|z(t)\|$  or   the squared magnitude $p(t) = \|z(t)\|^2$  are $($with  $\sigma = \sigma_{\rm R})$:
	:$$f_a(a) =		:$$f_a(a) =\left\{ \begin{array}{c} a/\sigma^2 \cdot {\rm e}^{ -{a^2}/(2\sigma^2)} \\0 \end{array} \right.\quad\begin{array}{*{1}c} {\rm for}\hspace{0.15cm} a \ge 0\\ {\rm for}\hspace{0.15cm} a < 0 \\ \end{array}\hspace{0.05cm},$$
	\left\{ \begin{array}{c} a/\sigma^2 \cdot {\rm e}^{ -{a^2}/(2\sigma^2)} \\		:$$f_p(p) =\left\{ \begin{array}{c} 1/(2\sigma^2) \cdot {\rm e}^{ -{p}/(2\sigma^2)} \\0 \end{array} \right.\quad\begin{array}{*{1}c} {\rm for}\hspace{0.15cm} p \ge 0\\ {\rm for}\hspace{0.15cm} p < 0 \\ \end{array}.$$
	0 \end{array} \right.\quad
	\begin{array}{*{1}c} {\rm for}\hspace{0.15cm} a \ge 0
	\\ {\rm for}\hspace{0.15cm} a < 0 \\ \end{array}
	\hspace{0.05cm},$$
	:$$f_p(p) =
	\left\{ \begin{array}{c} 1/(2\sigma^2) \cdot {\rm e}^{ -{p}/(2\sigma^2)} \\
	0 \end{array} \right.\quad
	\begin{array}{*{1}c} {\rm for}\hspace{0.15cm} p \ge 0
	\\ {\rm for}\hspace{0.15cm} p < 0 \\ \end{array}
	.$$
	* For channel  $\rm B$  ("Blue") only the point cloud is given.&nbsp It must be estimated whether Rayleigh fading is also present here, and if YES, how large the parameter  $\sigma = \sigma_{\rm B}$  is for this channel.		* For channel  $\rm B$  ("Blue") only the point cloud is given.&nbsp It must be estimated whether Rayleigh fading is also present here, and if YES, how large the parameter  $\sigma = \sigma_{\rm B}$  is for this channel.
	* Finally, subtask  '''(3)'''  also refers to the PDF  $f_{\it \phi}(\phi)$  of the phase function  $\phi(t)$ , which is defined as follows:		* Finally, subtask  '''(3)'''  also refers to the PDF  $f_{\it \phi}(\phi)$  of the phase function  $\phi(t)$ , which is defined as follows:
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	'''(2)'''  By measuring the two drawn circles you can see that for the "blue" channel the scattering of real– and imaginary part are larger by a factor of about  $1.4$  <br>$($exactly:  $\sqrt{2})$  than for the "red" channel:		'''(2)'''  By measuring the two drawn circles you can see that for the "blue" channel the scattering of real– and imaginary part are larger by a factor of about  $1.4$  <br>$($exactly:  $\sqrt{2})$  than for the "red" channel:
	$$\sigma_{\rm B} = \sigma_{\rm R} \cdot \sqrt{2} = 0.5 \cdot \sqrt{2}= {1}/{\sqrt{2}}\hspace{0.15cm} \underline{\approx 0.707}		$$\sigma_{\rm B} = \sigma_{\rm R} \cdot \sqrt{2} = 0.5 \cdot \sqrt{2}= {1}/{\sqrt{2}}\hspace{0.15cm} \underline{\approx 0.707}\hspace{0.05cm}.$$
	\hspace{0.05cm}.$$

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	* Finally, subtask  '''(3)'''  also refers to the PDF  $f_{\it \phi}(\phi)$  of the phase function  $\phi(t)$ , which is defined as follows:		* Finally, subtask  '''(3)'''  also refers to the PDF  $f_{\it \phi}(\phi)$  of the phase function  $\phi(t)$ , which is defined as follows:

	:$$\phi(t) = \arctan \hspace{0.15cm} \frac{y(t)}{x(t)}		:$$\phi(t) = \arctan \hspace{0.15cm} \frac{y(t)}{x(t)}\hspace{0.05cm}.$$
	\hspace{0.05cm}.$$


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	'''(7)'''  For the channel  $\rm R$  and $\sigma = 0.5$   ⇒   $\sigma^2 = 0.25$, we have		'''(7)'''  For the channel  $\rm R$  and $\sigma = 0.5$   ⇒   $\sigma^2 = 0.25$, we have
	:$${\rm Pr}(\|z(t)\| > 1) = {\rm e}^{-2} \hspace{0.15cm} \underline{\approx 0.135}		:$${\rm Pr}(\|z(t)\| > 1) = {\rm e}^{-2} \hspace{0.15cm} \underline{\approx 0.135}\hspace{0.05cm}.$$
	\hspace{0.05cm}.$$

	*In the upper graph this corresponds to the number of all points outside the circle drawn, based on the number  $N = 10,000$  of all points.		*In the upper graph this corresponds to the number of all points outside the circle drawn, based on the number  $N = 10,000$  of all points.

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	[[Category:Mobile Communications: Exercises\|^1.2 PDF of Rayleigh Fading^]]		[[Category:Mobile Communications: Exercises\|^1.2 PDF of Rayleigh Fading^]]
			[[de:Exercises:Exercise_1.3Z:_Rayleigh_Fading_Revisited]]

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	''Notes:''		''Notes:''
	* This task belongs to chapter  [[Mobile_Communications/Probability_density_of_Rayleigh_fading\|Probability density of Rayleigh fading]]  of this book.		* This task belongs to chapter  [[Mobile_Communications/Probability_density_of_Rayleigh_fading\|Probability density of Rayleigh fading]]  of this book.
	* A similar topic is treated with a different approach in chapter  [[Theory_of_Stochastic_Signals/Further_distributions\|Further distributions]]  of the book ~~&bdquo;~~Stochastic Signal Theory".		* A similar topic is treated with a different approach in chapter  [[Theory_of_Stochastic_Signals/Further_distributions\|Further distributions]]  of the book "Stochastic Signal Theory".
	* To check your results, you can use the interactive applet  [[Applets:PDF,_CDF_and_Moments_of_Special_Distributions\|PDF, CDF and Moments of Special Distributions]]  of the book ~~&bdquo;~~Stochastic Signal Theory".		* To check your results, you can use the interactive applet  [[Applets:PDF,_CDF_and_Moments_of_Special_Distributions\|PDF, CDF and Moments of Special Distributions]]  of the book "Stochastic Signal Theory".

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	''Notes:''		''Notes:''
	* This task belongs to chapter  [[Mobile_Communications/Probability_density_of_Rayleigh_fading\|Probability density of Rayleigh fading]]  of this book.		* This task belongs to chapter  [[Mobile_Communications/Probability_density_of_Rayleigh_fading\|Probability density of Rayleigh fading]]  of this book.
	* A similar topic is treated with a different approach in chapter  [[Theory_of_Stochastic_Signals/Further_distributions\|Further distributions]]  of the book &bdquo;Stochastic Signal Theory~~”~~.		* A similar topic is treated with a different approach in chapter  [[Theory_of_Stochastic_Signals/Further_distributions\|Further distributions]]  of the book &bdquo;Stochastic Signal Theory".
	* To check your results, you can use the interactive applet  [[Applets:PDF,_CDF_and_Moments_of_Special_Distributions\|PDF, CDF and Moments of Special Distributions]]  of the book &bdquo;Stochastic Signal Theory~~”~~.		* To check your results, you can use the interactive applet  [[Applets:PDF,_CDF_and_Moments_of_Special_Distributions\|PDF, CDF and Moments of Special Distributions]]  of the book &bdquo;Stochastic Signal Theory".

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	[[Category:~~Exercises for~~ Mobile Communications\|^1.2 PDF of Rayleigh Fading^]]		[[Category:Mobile Communications: Exercises\|^1.2 PDF of Rayleigh Fading^]]

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	</quiz>		</quiz>

	===~~Solutions~~===		===Solution===
	{{ML-Kopf}}		{{ML-Kopf}}
	'''(1)'''  <u>YES</u>:		'''(1)'''  <u>YES</u>:

Javier: Text replacement - "Mobile Kommunikation/Wahrscheinlichkeitsdichte des Rayleigh–Fadings" to "Mobile_Communications/Probability_Density_of_Rayleigh_Fading"

2020-12-15T13:56:33Z

Text replacement - "Mobile Kommunikation/Wahrscheinlichkeitsdichte des Rayleigh–Fadings" to "Mobile_Communications/Probability_Density_of_Rayleigh_Fading"

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	{{quiz-Header\|Buchseite=~~Mobile Kommunikation~~/~~Wahrscheinlichkeitsdichte des Rayleigh–Fadings~~}}		{{quiz-Header\|Buchseite=Mobile_Communications/Probability_Density_of_Rayleigh_Fading}}

	[[File:EN_Mob_A_1_3Z.png\|right\|frame\|Two channels, characterized by complex factor  $z(t)$]]		[[File:EN_Mob_A_1_3Z.png\|right\|frame\|Two channels, characterized by complex factor  $z(t)$]]

Guenter at 17:00, 4 December 2020

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	'''(5)'''  <u>Solution 3</u> is correct, as already explained in the sample solution for '''(4)'''.		'''(5)'''  <u>Solution 3</u> is correct, as already explained in the sample solution for  '''(4)'''.


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	'''(7)'''  For the channel  $\rm R$  and $\sigma = 0.5$, we have		'''(7)'''  For the channel  $\rm R$  and $\sigma = 0.5$   ⇒   $\sigma^2 = 0.25$, we have
	:$${\rm Pr}(\|z(t)\| > 1) = {\rm e}^{-2} \hspace{0.15cm} \underline{\approx 0.135}		:$${\rm Pr}(\|z(t)\| > 1) = {\rm e}^{-2} \hspace{0.15cm} \underline{\approx 0.135}
	\hspace{0.05cm}.$$		\hspace{0.05cm}.$$