Aufgaben:Exercise 4.17Z: Rayleigh and Rice Distribution - Revision history

Maintenance script: Fix interlanguage link: resolve redirect chain

2026-03-16T15:56:27Z

Fix interlanguage link: resolve redirect chain

← Older revision		Revision as of 17:56, 16 March 2026
Line 104:		Line 104:

	[[Category:Digital Signal Transmission: Exercises\|^4.5 Non-Coherent Demodulation^]]		[[Category:Digital Signal Transmission: Exercises\|^4.5 Non-Coherent Demodulation^]]
	[[de:4.~~17Z_Rayleigh~~-~~_und_Riceverteilung~~]]		[[de:Aufgaben:Aufgabe 4.17Z: Rayleigh- und Riceverteilung]]

Maintenance script: Join split $$ formula lines for correct rendering

2026-03-16T13:31:06Z

Join split $$ formula lines for correct rendering

← Older revision		Revision as of 15:31, 16 March 2026
Line 5:		Line 5:
	For the study of transmission systems,  the Rayleigh and Rice distributions are of great importance.  In the following let  $y$  be a Rayleigh or a Rice distributed random variable and  $\eta$  in each case a realization of it.		For the study of transmission systems,  the Rayleigh and Rice distributions are of great importance.  In the following let  $y$  be a Rayleigh or a Rice distributed random variable and  $\eta$  in each case a realization of it.
	* The  "Rayleigh distribution"  results thereby for the probability density function  $\rm (PDF)$  of a random variable  $y$,  which results from the two Gaussian distributed and statistically independent components  $u$  and  $v$  $($both with the standard deviation  $\sigma_n)$  as follows:		* The  "Rayleigh distribution"  results thereby for the probability density function  $\rm (PDF)$  of a random variable  $y$,  which results from the two Gaussian distributed and statistically independent components  $u$  and  $v$  $($both with the standard deviation  $\sigma_n)$  as follows:
	:$$y = \sqrt{u^2 + v^2} \hspace{0.1cm} \Rightarrow \hspace{0.1cm} p_y (\eta) = \frac{\eta}{\sigma_n^2}		:$$y = \sqrt{u^2 + v^2} \hspace{0.1cm} \Rightarrow \hspace{0.1cm} p_y (\eta) = \frac{\eta}{\sigma_n^2}\cdot {\rm exp } \left [ - \frac{\eta^2}{2 \sigma_n^2}\right ]\hspace{0.01cm}.$$
	\cdot {\rm exp } \left [ - \frac{\eta^2}{2 \sigma_n^2}\right ]
	\hspace{0.01cm}.$$

	* The  "Rice distribution"  is obtained under otherwise identical boundary conditions for the application case where a constant  $C$  is still added to one of the two components,  for example:		* The  "Rice distribution"  is obtained under otherwise identical boundary conditions for the application case where a constant  $C$  is still added to one of the two components,  for example:
	:$$y = \sqrt{(u+C)^2 + v^2} \hspace{0.3cm} \Rightarrow \hspace{0.3cm} p_y (\eta) = \frac{\eta}{\sigma_n^2}		:$$y = \sqrt{(u+C)^2 + v^2} \hspace{0.3cm} \Rightarrow \hspace{0.3cm} p_y (\eta) = \frac{\eta}{\sigma_n^2}\cdot {\rm exp } \left [ - \frac{\eta^2 + C^2}{2 \sigma_n^2}\right ] \cdot {\rm I }_0 \left [ \frac{\eta \cdot C}{ \sigma_n^2}\right ]\hspace{0.05cm}.$$
	\cdot {\rm exp } \left [ - \frac{\eta^2 + C^2}{2 \sigma_n^2}\right ] \cdot {\rm I }_0 \left [ \frac{\eta \cdot C}{ \sigma_n^2}\right ]
	\hspace{0.05cm}.$$

	In this equation,  ${\rm I}_0(x)$  denotes the  [[Digital_Signal_Transmission/Carrier_Frequency_Systems_with_Non-Coherent_Demodulation#Rayleigh_and_Rice_Distribution\|"modified zero-order Bessel function"]].		In this equation,  ${\rm I}_0(x)$  denotes the  [[Digital_Signal_Transmission/Carrier_Frequency_Systems_with_Non-Coherent_Demodulation#Rayleigh_and_Rice_Distribution\|"modified zero-order Bessel function"]].
Line 32:		Line 28:

	* For the standard deviation  (root of the variance)  of the Rayleigh distribution holds:		* For the standard deviation  (root of the variance)  of the Rayleigh distribution holds:
	:$$\sigma_y = \sigma_n \cdot \sqrt{2 - {\pi}/{2 }} \hspace{0.2cm} \approx \hspace{0.2cm} 0.655 \cdot \sigma_n		:$$\sigma_y = \sigma_n \cdot \sqrt{2 - {\pi}/{2 }} \hspace{0.2cm} \approx \hspace{0.2cm} 0.655 \cdot \sigma_n\hspace{0.05cm}.$$
	\hspace{0.05cm}.$$

	* For the standard deviation or for the variance of the Rice distribution in general only a complicated expression with hypergeometric functions can be given,  otherwise only an approximation for  $C \gg \sigma_n$  corresponding to the Gaussian distribution.		* For the standard deviation or for the variance of the Rice distribution in general only a complicated expression with hypergeometric functions can be given,  otherwise only an approximation for  $C \gg \sigma_n$  corresponding to the Gaussian distribution.
Line 46:		Line 41:

	* Given is also the following indefinite integral:		* Given is also the following indefinite integral:
	:$$\int x \cdot {\rm e }^{-x^2} \,{\rm d} x =		:$$\int x \cdot {\rm e }^{-x^2} \,{\rm d} x =-{1}/{2} \cdot {\rm e }^{-x^2} + {\rm const. } $$
	-{1}/{2} \cdot {\rm e }^{-x^2} + {\rm const. } $$


Line 85:		Line 79:
	'''(2)'''  You can see from the graph:  The mean value of the Gaussian distribution is  $\underline {C = 4}$ and the standard deviation is $\underline {\sigma_n = 1}$.		'''(2)'''  You can see from the graph:  The mean value of the Gaussian distribution is  $\underline {C = 4}$ and the standard deviation is $\underline {\sigma_n = 1}$.
	*It was given that  $C$  and  $\sigma_n$  were integers.  Thus the two density functions are:		*It was given that  $C$  and  $\sigma_n$  were integers.  Thus the two density functions are:
	:$$p_{\rm I} (\eta) \hspace{-0.1cm} \ = \ \hspace{-0.1cm} {\eta}		:$$p_{\rm I} (\eta) \hspace{-0.1cm} \ = \ \hspace{-0.1cm} {\eta}\cdot {\rm exp } \left [ - \frac{\eta^2 + 16}{2 }\right ] \cdot {\rm I }_0 (4\eta ) \approx\frac{1}{\sqrt{2\pi }}\cdot {\rm exp } \left [ - \frac{(\eta-4)^2 }{2 }\right ]\hspace{0.05cm},$$
	\cdot {\rm exp } \left [ - \frac{\eta^2 + 16}{2 }\right ] \cdot {\rm I }_0 (4\eta ) \approx		:$$ p_{\rm II} (\eta) \hspace{-0.1cm} \ = \ \hspace{-0.1cm} {\eta}\cdot {\rm exp } \left [ - \frac{\eta^2 }{2 }\right ] \hspace{0.05cm}.$$
	\frac{1}{\sqrt{2\pi }}\cdot {\rm exp } \left [ - \frac{(\eta-4)^2 }{2 }\right ]\hspace{0.05cm},$$
	:$$ p_{\rm II} (\eta) \hspace{-0.1cm} \ = \ \hspace{-0.1cm} {\eta}
	\cdot {\rm exp } \left [ - \frac{\eta^2 }{2 }\right ] \hspace{0.05cm}.$$


	'''(3)'''  <u>Solution 2</u>  is correct,  as can already be seen from the graph.  A calculation confirms this result:		'''(3)'''  <u>Solution 2</u>  is correct,  as can already be seen from the graph.  A calculation confirms this result:
	:$$\sigma_{\rm Rice}^2 \hspace{-0.1cm} \ = \ \hspace{-0.1cm} \sigma_n^2 = 1\hspace{0.05cm},$$		:$$\sigma_{\rm Rice}^2 \hspace{-0.1cm} \ = \ \hspace{-0.1cm} \sigma_n^2 = 1\hspace{0.05cm},$$
	:$$ \sigma_{\rm Rayl}^2 \hspace{-0.1cm} \ = \ \hspace{-0.1cm} \sigma_n^2 \cdot ({2 - {\pi}/{2 }}) \approx 0.429		:$$ \sigma_{\rm Rayl}^2 \hspace{-0.1cm} \ = \ \hspace{-0.1cm} \sigma_n^2 \cdot ({2 - {\pi}/{2 }}) \approx 0.429\hspace{0.05cm}.$$
	\hspace{0.05cm}.$$


	'''(4)'''  In general,  the probability that  $y$  is greater than a certain value  $y_0$  is equal to		'''(4)'''  In general,  the probability that  $y$  is greater than a certain value  $y_0$  is equal to
	:$${\rm Pr}(y > y_0) = \int_{y_0}^{\infty} \frac{\eta}{\sigma_n^2}		:$${\rm Pr}(y > y_0) = \int_{y_0}^{\infty} \frac{\eta}{\sigma_n^2}\cdot {\rm exp } \left [ - \frac{\eta^2 }{2 \sigma_n^2}\right ] \,{\rm d} \eta\hspace{0.05cm}.$$
	\cdot {\rm exp } \left [ - \frac{\eta^2 }{2 \sigma_n^2}\right ] \,{\rm d} \eta
	\hspace{0.05cm}.$$

	*With the substitution  $x^2 = \eta^2/(2\sigma_n^2)$  can be written for this:		*With the substitution  $x^2 = \eta^2/(2\sigma_n^2)$  can be written for this:
	:$${\rm Pr}(y > y_0) = 2 \cdot \hspace{-0.05cm}\int_{y_0/(\sqrt{2}\hspace{0.03cm} \cdot \hspace{0.03cm} \sigma_n)}^{\infty} \hspace{-0.5cm}x		:$${\rm Pr}(y > y_0) = 2 \cdot \hspace{-0.05cm}\int_{y_0/(\sqrt{2}\hspace{0.03cm} \cdot \hspace{0.03cm} \sigma_n)}^{\infty} \hspace{-0.5cm}x\cdot {\rm e }^{ - x^2} \,{\rm d} x = \left [{\rm e }^{ - x^2} \right ]_{\sqrt{2}\hspace{0.03cm} \cdot \hspace{0.03cm} \sigma_n}^{\infty}= {\rm exp } \left [ -\frac{ y_0^2 }{2 \sigma_n^2 }\right ]\hspace{0.05cm}.$$
	\cdot {\rm e }^{ - x^2} \,{\rm d} x = \left [{\rm e }^{ - x^2} \right ]_{\sqrt{2}\hspace{0.03cm} \cdot \hspace{0.03cm} \sigma_n}^{\infty}
	= {\rm exp } \left [ -\frac{ y_0^2 }{2 \sigma_n^2 }\right ]\hspace{0.05cm}.$$

	*Here the indefinite integral given in the front was used.  In particular:		*Here the indefinite integral given in the front was used.  In particular:

Maintenance script: Add German interlanguage link

2026-03-16T12:25:50Z

Add German interlanguage link

← Older revision		Revision as of 14:25, 16 March 2026
Line 118:		Line 118:

	[[Category:Digital Signal Transmission: Exercises\|^4.5 Non-Coherent Demodulation^]]		[[Category:Digital Signal Transmission: Exercises\|^4.5 Non-Coherent Demodulation^]]
			[[de:4.17Z_Rayleigh-_und_Riceverteilung]]

Guenter at 11:41, 10 October 2022

2022-10-10T11:41:36Z

← Older revision		Revision as of 13:41, 10 October 2022
Line 2:		Line 2:
	{{quiz-Header\|Buchseite=Digital_Signal_Transmission/Carrier_Frequency_Systems_with_Non-Coherent_Demodulation}}		{{quiz-Header\|Buchseite=Digital_Signal_Transmission/Carrier_Frequency_Systems_with_Non-Coherent_Demodulation}}

	[[File:~~P_ID2079__Dig_Z_4_17~~.png\|right\|frame\|Rice (top) and Rayleigh (bottom)]]		[[File:P_ID2079__Dig_Z_4_17_ret.png\|right\|frame\|Rice (top) and Rayleigh (bottom)]]
	For the study of transmission systems,  the Rayleigh and Rice distributions are of great importance.  In the following let  $y$  be a Rayleigh or a Rice distributed random variable and  $\eta$  in each case a realization of it.		For the study of transmission systems,  the Rayleigh and Rice distributions are of great importance.  In the following let  $y$  be a Rayleigh or a Rice distributed random variable and  $\eta$  in each case a realization of it.
	* The  "Rayleigh distribution"  results thereby for the probability density function  $\rm (PDF)$  of a random variable  $y$,  which results from the two Gaussian distributed and statistically independent components  $u$  and  $v$  $($both with the standard deviation  $\sigma_n)$  as follows:		* The  "Rayleigh distribution"  results thereby for the probability density function  $\rm (PDF)$  of a random variable  $y$,  which results from the two Gaussian distributed and statistically independent components  $u$  and  $v$  $($both with the standard deviation  $\sigma_n)$  as follows:

Guenter at 11:54, 30 August 2022

2022-08-30T11:54:27Z

← Older revision		Revision as of 13:54, 30 August 2022
Line 78:		Line 78:
	===Solution===		===Solution===
	{{ML-Kopf}}		{{ML-Kopf}}
	'''(1)'''  The <u>second solution</u> is correct:		'''(1)'''  The  <u>second solution</u>  is correct:
	*The upper graph shows approximately a Gaussian distribution and belongs accordingly to the Rice distribution.		*The upper graph shows approximately a Gaussian distribution and belongs accordingly to the Rice distribution.



	'''(2)'''  You can see from the graph: The mean value of the Gaussian distribution is $\underline {C = 4}$ and the standard deviation is $\underline {\sigma_n = 1}$.		'''(2)'''  You can see from the graph:  The mean value of the Gaussian distribution is  $\underline {C = 4}$ and the standard deviation is $\underline {\sigma_n = 1}$.
	*It was given that $C$ and $\sigma_n$ were integers.		*It was given that  $C$  and  $\sigma_n$  were integers.  Thus the two density functions are:
	*Thus the two density functions are:
	:$$p_{\rm I} (\eta) \hspace{-0.1cm} \ = \ \hspace{-0.1cm} {\eta}		:$$p_{\rm I} (\eta) \hspace{-0.1cm} \ = \ \hspace{-0.1cm} {\eta}
	\cdot {\rm exp } \left [ - \frac{\eta^2 + 16}{2 }\right ] \cdot {\rm I }_0 (4\eta ) \approx		\cdot {\rm exp } \left [ - \frac{\eta^2 + 16}{2 }\right ] \cdot {\rm I }_0 (4\eta ) \approx
Line 93:		Line 92:


	'''(3)'''  <u>Solution 2</u> is correct, as can already be seen from the graph. A calculation confirms this result:		'''(3)'''  <u>Solution 2</u>  is correct,  as can already be seen from the graph.  A calculation confirms this result:
	:$$\sigma_{\rm Rice}^2 \hspace{-0.1cm} \ = \ \hspace{-0.1cm} \sigma_n^2 = 1\hspace{0.05cm},$$		:$$\sigma_{\rm Rice}^2 \hspace{-0.1cm} \ = \ \hspace{-0.1cm} \sigma_n^2 = 1\hspace{0.05cm},$$
	:$$ \sigma_{\rm Rayl}^2 \hspace{-0.1cm} \ = \ \hspace{-0.1cm} \sigma_n^2 \cdot ({2 - {\pi}/{2 }}) \approx 0.429		:$$ \sigma_{\rm Rayl}^2 \hspace{-0.1cm} \ = \ \hspace{-0.1cm} \sigma_n^2 \cdot ({2 - {\pi}/{2 }}) \approx 0.429
Line 99:		Line 98:


	'''(4)'''  In general, the probability that $y$ is greater than a value $y_0$ is equal to		'''(4)'''  In general,  the probability that  $y$  is greater than a certain value  $y_0$  is equal to
	:$${\rm Pr}(y > y_0) = \int_{y_0}^{\infty} \frac{\eta}{\sigma_n^2}		:$${\rm Pr}(y > y_0) = \int_{y_0}^{\infty} \frac{\eta}{\sigma_n^2}
	\cdot {\rm exp } \left [ - \frac{\eta^2 }{2 \sigma_n^2}\right ] \,{\rm d} \eta		\cdot {\rm exp } \left [ - \frac{\eta^2 }{2 \sigma_n^2}\right ] \,{\rm d} \eta
	\hspace{0.05cm}.$$		\hspace{0.05cm}.$$

	*With the substitution $x^2 = \eta^2/(2\sigma_n^2)$ can be written for this:		*With the substitution  $x^2 = \eta^2/(2\sigma_n^2)$  can be written for this:
	:$${\rm Pr}(y > y_0) = 2 \cdot \hspace{-0.05cm}\int_{y_0/(\sqrt{2}\hspace{0.03cm} \cdot \hspace{0.03cm} \sigma_n)}^{\infty} \hspace{-0.5cm}x		:$${\rm Pr}(y > y_0) = 2 \cdot \hspace{-0.05cm}\int_{y_0/(\sqrt{2}\hspace{0.03cm} \cdot \hspace{0.03cm} \sigma_n)}^{\infty} \hspace{-0.5cm}x
	\cdot {\rm e }^{ - x^2} \,{\rm d} x = \left [{\rm e }^{ - x^2} \right ]_{\sqrt{2}\hspace{0.03cm} \cdot \hspace{0.03cm} \sigma_n}^{\infty}		\cdot {\rm e }^{ - x^2} \,{\rm d} x = \left [{\rm e }^{ - x^2} \right ]_{\sqrt{2}\hspace{0.03cm} \cdot \hspace{0.03cm} \sigma_n}^{\infty}
	= {\rm exp } \left [ -\frac{ y_0^2 }{2 \sigma_n^2 }\right ]\hspace{0.05cm}.$$		= {\rm exp } \left [ -\frac{ y_0^2 }{2 \sigma_n^2 }\right ]\hspace{0.05cm}.$$

	*Here the indefinite integral given in the front was used. In particular:		*Here the indefinite integral given in the front was used.  In particular:
	:$${\rm Pr}(y > \sigma_n) \hspace{-0.1cm} \ = \ \hspace{-0.1cm} {\rm e }^{ - 0.5} \hspace{0.15cm} \underline{\approx 60.7 \%} \hspace{0.05cm},$$		:$${\rm Pr}(y > \sigma_n) \hspace{-0.1cm} \ = \ \hspace{-0.1cm} {\rm e }^{ - 0.5} \hspace{0.15cm} \underline{\approx 60.7 \%} \hspace{0.05cm},$$
	:$$ {\rm Pr}(y > 2\sigma_n) \hspace{-0.1cm} \ = \ \hspace{-0.1cm} {\rm e }^{ - 2.0} \hspace{0.15cm} \underline{\approx 13.5 \%} \hspace{0.05cm},$$		:$$ {\rm Pr}(y > 2\sigma_n) \hspace{-0.1cm} \ = \ \hspace{-0.1cm} {\rm e }^{ - 2.0} \hspace{0.15cm} \underline{\approx 13.5 \%} \hspace{0.05cm},$$

Guenter at 11:48, 30 August 2022

2022-08-30T11:48:50Z

Hwang at 14:33, 16 August 2022

2022-08-16T14:33:09Z

Show changes

Guenter at 15:27, 15 August 2022

2022-08-15T15:27:00Z

← Older revision		Revision as of 17:27, 15 August 2022
Line 114:		Line 114:


	[[Category:Digital Signal Transmission: Exercises\|^4.5 ~~Inkohärente~~ Demodulation^]]		[[Category:Digital Signal Transmission: Exercises\|^4.5 Non-Coherent Demodulation^]]

Guenter: Guenter moved page Aufgaben:Aufgabe 4.17Z: Rayleigh- und Riceverteilung to Aufgaben:Exercise 4.17Z: Rayleigh and Rice Distribution

2022-08-15T15:20:51Z

Guenter moved page Aufgaben:Aufgabe 4.17Z: Rayleigh- und Riceverteilung to Aufgaben:Exercise 4.17Z: Rayleigh and Rice Distribution

← Older revision	Revision as of 17:20, 15 August 2022
(No difference)

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

2021-03-23T12:51:03Z

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

← Older revision		Revision as of 14:51, 23 March 2021
Line 114:		Line 114:


	[[Category:~~Aufgaben zu Digitalsignalübertragung~~\|^4.5 Inkohärente Demodulation^]]		[[Category:Digital Signal Transmission: Exercises\|^4.5 Inkohärente Demodulation^]]

← Older revision		Revision as of 13:48, 30 August 2022
Line 3:		Line 3:

	[[File:P_ID2079__Dig_Z_4_17.png\|right\|frame\|Rice (top) and Rayleigh (bottom)]]		[[File:P_ID2079__Dig_Z_4_17.png\|right\|frame\|Rice (top) and Rayleigh (bottom)]]
	For the study of ~~message~~ systems, the Rayleigh and Rice distributions are of great importance. In the following let  $y$  be a Rayleigh or a Rice distributed random variable and  $\eta$  in each case a realization of it.		For the study of transmission systems,  the Rayleigh and Rice distributions are of great importance.  In the following let  $y$  be a Rayleigh or a Rice distributed random variable and  $\eta$  in each case a realization of it.
	* The ~~<i>~~Rayleigh distribution~~</i>~~  results thereby for the probability density function (~~short:~~ PDF) of a random variable  $y$, which results from the two Gaussian distributed and statistically independent components  $u$  and  $v$  $($both with the standard deviation  $\sigma_n)$  as follows:		* The  "Rayleigh distribution"  results thereby for the probability density function  $\rm (PDF)$  of a random variable  $y$,  which results from the two Gaussian distributed and statistically independent components  $u$  and  $v$  $($both with the standard deviation  $\sigma_n)$  as follows:
	:$$y = \sqrt{u^2 + v^2} \hspace{0.1cm} \Rightarrow \hspace{0.1cm} p_y (\eta) = \frac{\eta}{\sigma_n^2}		:$$y = \sqrt{u^2 + v^2} \hspace{0.1cm} \Rightarrow \hspace{0.1cm} p_y (\eta) = \frac{\eta}{\sigma_n^2}
	\cdot {\rm exp } \left [ - \frac{\eta^2}{2 \sigma_n^2}\right ]		\cdot {\rm exp } \left [ - \frac{\eta^2}{2 \sigma_n^2}\right ]
	\hspace{0.01cm}.$$		\hspace{0.01cm}.$$

	* The ~~<i>~~Rice distribution~~</i>~~  is obtained under otherwise identical boundary conditions for the application case where a constant  $C$  is still added to one of the two components, for example:		* The  "Rice distribution"  is obtained under otherwise identical boundary conditions for the application case where a constant  $C$  is still added to one of the two components,  for example:
	:$$y = \sqrt{(u+C)^2 + v^2} \hspace{0.3cm} \Rightarrow \hspace{0.3cm} p_y (\eta) = \frac{\eta}{\sigma_n^2}		:$$y = \sqrt{(u+C)^2 + v^2} \hspace{0.3cm} \Rightarrow \hspace{0.3cm} p_y (\eta) = \frac{\eta}{\sigma_n^2}
	\cdot {\rm exp } \left [ - \frac{\eta^2 + C^2}{2 \sigma_n^2}\right ] \cdot {\rm I }_0 \left [ \frac{\eta \cdot C}{ \sigma_n^2}\right ]		\cdot {\rm exp } \left [ - \frac{\eta^2 + C^2}{2 \sigma_n^2}\right ] \cdot {\rm I }_0 \left [ \frac{\eta \cdot C}{ \sigma_n^2}\right ]
Line 16:		Line 16:
	In this equation,  ${\rm I}_0(x)$  denotes the  [[Digital_Signal_Transmission/Carrier_Frequency_Systems_with_Non-Coherent_Demodulation#Rayleigh_and_Rice_Distribution\|"modified zero-order Bessel function"]].		In this equation,  ${\rm I}_0(x)$  denotes the  [[Digital_Signal_Transmission/Carrier_Frequency_Systems_with_Non-Coherent_Demodulation#Rayleigh_and_Rice_Distribution\|"modified zero-order Bessel function"]].

	In the graph, the two density functions are shown, but it is not indicated whether  $p_{\hspace{0.03cm}\rm I}(\eta)$  or  $p_{\hspace{0.03cm}\rm II}(\eta)$  belong to a Rayleigh or a Rice distribution, respectively.		In the graph,  the two probability density functions are shown,  but it is not indicated whether  $p_{\hspace{0.03cm}\rm I}(\eta)$  or  $p_{\hspace{0.03cm}\rm II}(\eta)$  belong to a Rayleigh or a Rice distribution,  respectively.
	* It is only known that one Rayleigh and one Rice distribution is shown.		* It is only known that one Rayleigh and one Rice distribution is shown.

	* The parameter  $\sigma_n$  is the same for both distributions.		* The parameter  $\sigma_n$  is the same for both distributions.


	For your decision whether to assign  $p_{\hspace{0.03cm}\rm I}(\eta)$  or  $p_{\rm II}(\hspace{0.03cm}\eta)$  to the Rice distribution and for the determination of the PDF parameters you can consider the following statements:		For your decision whether to assign  $p_{\hspace{0.03cm}\rm I}(\eta)$  or  $p_{\rm II}(\hspace{0.03cm}\eta)$  to the Rice distribution and for the determination of the PDF parameters you can consider the following statements:
	* For large values of the quotient  $C/\sigma_n$,  the Rice distribution can be approximated by a Gaussian distribution with mean  $C$  and standard deviation  $\sigma_n$.		* For large values of the quotient  $C/\sigma_n$,  the Rice distribution can be approximated by a Gaussian distribution with mean  $C$  and standard deviation  $\sigma_n$.

	* The values of  $C$  and  $\sigma_n$  underlying the graph are integers.		* The values of  $C$  and  $\sigma_n$  underlying the graph are integers.


	Regarding the Rayleigh distribution, note:		Regarding the Rayleigh distribution,  note:
	* The same  $\sigma_n$  is used for both distributions.		* The same  $\sigma_n$  is used for both distributions.
	* For the standard deviation (root of the variance) of the Rayleigh distribution holds:
			* For the standard deviation  (root of the variance)  of the Rayleigh distribution holds:
	:$$\sigma_y = \sigma_n \cdot \sqrt{2 - {\pi}/{2 }} \hspace{0.2cm} \approx \hspace{0.2cm} 0.655 \cdot \sigma_n		:$$\sigma_y = \sigma_n \cdot \sqrt{2 - {\pi}/{2 }} \hspace{0.2cm} \approx \hspace{0.2cm} 0.655 \cdot \sigma_n
	\hspace{0.05cm}.$$		\hspace{0.05cm}.$$
	* For the standard deviation or for the variance of the Rice distribution in general only a complicated expression with hypergeometric functions can be given, otherwise only an approximation for  $C \gg \sigma_n$  corresponding to the Gaussian distribution.
			* For the standard deviation or for the variance of the Rice distribution in general only a complicated expression with hypergeometric functions can be given,  otherwise only an approximation for  $C \gg \sigma_n$  corresponding to the Gaussian distribution.


Line 38:		Line 42:


	''Notes:''		Notes:
	* This exercise belongs to the chapter  [[Digital_Signal_Transmission/Carrier_Frequency_Systems_with_Non-Coherent_Demodulation\|"Carrier Frequency Systems with Non-Coherent Demodulation"]].		* This exercise belongs to the chapter  [[Digital_Signal_Transmission/Carrier_Frequency_Systems_with_Non-Coherent_Demodulation\|"Carrier Frequency Systems with Non-Coherent Demodulation"]].