Aufgaben:Exercise 1.4: Rayleigh PDF and Jakes PDS - Revision history

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

2026-03-16T13:47:31Z

Join split $$ formula lines for correct rendering

← Older revision		Revision as of 15:47, 16 March 2026
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	[[File:P_ID2119__Mob_A_1_4.png\|right\|frame\| PDF and  $\|z(t)\|$  for Rayleigh fading with Doppler effect]]		[[File:P_ID2119__Mob_A_1_4.png\|right\|frame\| PDF and  $\|z(t)\|$  for Rayleigh fading with Doppler effect]]
	We consider two different mobile radio channels with  [[Mobile_Communications/Wahrscheinlichkeitsdichte_des_Rayleigh%E2%80%93Fadings#Beispielhafte_Signalverl.C3.A4ufe_bei_Rayleigh.E2.80.93Fading\|Rayleigh fading]]. In both cases the PDF of the magnitude  $a(t) = \|z(t)\| ≥ 0$  is		We consider two different mobile radio channels with  [[Mobile_Communications/Wahrscheinlichkeitsdichte_des_Rayleigh%E2%80%93Fadings#Beispielhafte_Signalverl.C3.A4ufe_bei_Rayleigh.E2.80.93Fading\|Rayleigh fading]]. In both cases the PDF of the magnitude  $a(t) = \|z(t)\| ≥ 0$  is
	:$$f_a(a) = \frac{a}{\sigma^2} \cdot {\rm e}^{ -{a^2}/(2\sigma^2)}		:$$f_a(a) = \frac{a}{\sigma^2} \cdot {\rm e}^{ -{a^2}/(2\sigma^2)}\hspace{0.05cm}.$$
	\hspace{0.05cm}.$$

	The probability that this magnitude is not greater than a given value  $A$  is		The probability that this magnitude is not greater than a given value  $A$  is
	:$${\rm Pr}(\|z(t)\| \le A) = 1 - {\rm e}^{ -{A^2}/(2\sigma^2)}		:$${\rm Pr}(\|z(t)\| \le A) = 1 - {\rm e}^{ -{A^2}/(2\sigma^2)}\hspace{0.05cm}.$$
	\hspace{0.05cm}.$$

	The two channels, which are designated according to the colors "Red" and "Blue" in the graphs with  $\rm R$  and  $\rm B$  respectively, differ in the speed  $v$  and thus in the form of the power-spectral density  $\rm (PSD)$   ${\it \Phi}_z(f_{\rm D})$.		The two channels, which are designated according to the colors "Red" and "Blue" in the graphs with  $\rm R$  and  $\rm B$  respectively, differ in the speed  $v$  and thus in the form of the power-spectral density  $\rm (PSD)$   ${\it \Phi}_z(f_{\rm D})$.
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	*For a Doppler frequency  $f_{\rm D}$  with  $\|f_{\rm D}\| <f_{\rm D,\hspace{0.1cm}max}$  the Jakes spectrum is given by		*For a Doppler frequency  $f_{\rm D}$  with  $\|f_{\rm D}\| <f_{\rm D,\hspace{0.1cm}max}$  the Jakes spectrum is given by
	:$${\it \Phi}_z(f_{\rm D}) = \frac{1}{\pi \hspace{-0.05cm}\cdot \hspace{-0.05cm}f_{\rm D, \hspace{0.1cm} max} \hspace{-0.05cm}\cdot \hspace{-0.05cm}\sqrt{ 1 \hspace{-0.05cm}- \hspace{-0.05cm}(f_{\rm D}/f_{\rm D, \hspace{0.1cm} max})^2} }		:$${\it \Phi}_z(f_{\rm D}) = \frac{1}{\pi \hspace{-0.05cm}\cdot \hspace{-0.05cm}f_{\rm D, \hspace{0.1cm} max} \hspace{-0.05cm}\cdot \hspace{-0.05cm}\sqrt{ 1 \hspace{-0.05cm}- \hspace{-0.05cm}(f_{\rm D}/f_{\rm D, \hspace{0.1cm} max})^2} }\hspace{0.05cm}.$$
	\hspace{0.05cm}.$$

	*For Doppler frequencies outside this interval from  $-f_{\rm D,\hspace{0.1cm}max}$  to  $+f_{\rm D,\hspace{0.1cm}max}$,   we have ${\it \Phi}_z(f_{\rm D})=0$.		*For Doppler frequencies outside this interval from  $-f_{\rm D,\hspace{0.1cm}max}$  to  $+f_{\rm D,\hspace{0.1cm}max}$,   we have ${\it \Phi}_z(f_{\rm D})=0$.
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	*The Rayleigh PDF and its derivative are		*The Rayleigh PDF and its derivative are
	:$$f_a(a) \hspace{-0.1cm} = \hspace{-0.1cm} \frac{a}{\sigma^2} \cdot {\rm e}^{ -a^2/(2\sigma^2)} \hspace{0.05cm},$$		:$$f_a(a) \hspace{-0.1cm} = \hspace{-0.1cm} \frac{a}{\sigma^2} \cdot {\rm e}^{ -a^2/(2\sigma^2)} \hspace{0.05cm},$$
	:$$\frac{{\rm d}f_a(a)}{{\rm d}a} \hspace{-0.1cm} = \hspace{-0.1cm} \frac{1}{\sigma^2} \cdot {\rm e}^{ -a^2/(2\sigma^2)}-		:$$\frac{{\rm d}f_a(a)}{{\rm d}a} \hspace{-0.1cm} = \hspace{-0.1cm} \frac{1}{\sigma^2} \cdot {\rm e}^{ -a^2/(2\sigma^2)}-\frac{a^2}{\sigma^4} \cdot {\rm e}^{ -a^2/(2\sigma^2)} \hspace{0.05cm}.$$
	\frac{a^2}{\sigma^4} \cdot {\rm e}^{ -a^2/(2\sigma^2)} \hspace{0.05cm}.$$

	*By setting the derivative to  $0$, you can show that the maximum of the PDF occurs at  $a = \sigma$.  Since the Rayleigh PDF applies to both channels, it follows that		*By setting the derivative to  $0$, you can show that the maximum of the PDF occurs at  $a = \sigma$.  Since the Rayleigh PDF applies to both channels, it follows that
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	'''(2)'''  As they fading coefficients have the same PDF, the desired probability is also the same for both channels.		'''(2)'''  As they fading coefficients have the same PDF, the desired probability is also the same for both channels.
	*Using the given equation, we have		*Using the given equation, we have
	:$${\rm Pr}(a \le 0.316) = {\rm Pr}(20 \cdot {\rm lg}\hspace{0.15cm} a \le -10\,\,{\rm dB}) = 1 - {\rm e}^{ -{0.316^2}/(2\sigma^2)}		:$${\rm Pr}(a \le 0.316) = {\rm Pr}(20 \cdot {\rm lg}\hspace{0.15cm} a \le -10\,\,{\rm dB}) = 1 - {\rm e}^{ -{0.316^2}/(2\sigma^2)}= 1- 0.951 \hspace{0.15cm} \underline{ \approx 4.9 \%}\hspace{0.05cm}.$$
	= 1- 0.951 \hspace{0.15cm} \underline{ \approx 4.9 \%}
	\hspace{0.05cm}.$$

Hwang at 06:05, 18 September 2022

2022-09-18T06:05:01Z

← Older revision		Revision as of 08:05, 18 September 2022
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	\hspace{0.05cm}.$$		\hspace{0.05cm}.$$

	The probability that this ~~amount~~ is not greater than a given value  $A$  is		The probability that this magnitude is not greater than a given value  $A$  is
	:$${\rm Pr}(\|z(t)\| \le A) = 1 - {\rm e}^{ -{A^2}/(2\sigma^2)}		:$${\rm Pr}(\|z(t)\| \le A) = 1 - {\rm e}^{ -{A^2}/(2\sigma^2)}
	\hspace{0.05cm}.$$		\hspace{0.05cm}.$$

Bene: Text replacement - "power spectral density" to "power-spectral density"

2022-02-17T11:41:31Z

Text replacement - "power spectral density" to "power-spectral density"

← Older revision		Revision as of 13:41, 17 February 2022
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	\hspace{0.05cm}.$$		\hspace{0.05cm}.$$

	The two channels, which are designated according to the colors "Red" and "Blue" in the graphs with  $\rm R$  and  $\rm B$  respectively, differ in the speed  $v$  and thus in the form of the power spectral density  $\rm (PSD)$   ${\it \Phi}_z(f_{\rm D})$.		The two channels, which are designated according to the colors "Red" and "Blue" in the graphs with  $\rm R$  and  $\rm B$  respectively, differ in the speed  $v$  and thus in the form of the power-spectral density  $\rm (PSD)$   ${\it \Phi}_z(f_{\rm D})$.

	*In both cases, however, the PDS is a [[Mobile_Communications/Statistical_bindings_within_the_Rayleigh_process\|Jakes spectrum]].		*In both cases, however, the PDS is a [[Mobile_Communications/Statistical_bindings_within_the_Rayleigh_process\|Jakes spectrum]].

Bene: Text replacement - "autocorrelation" to "auto-correlation"

2022-02-17T11:29:04Z

Text replacement - "autocorrelation" to "auto-correlation"

← Older revision		Revision as of 13:29, 17 February 2022
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	The corresponding descriptor in the time domain is the ~~autocorrelation~~ function  $\rm (ACF)$:		The corresponding descriptor in the time domain is the auto-correlation function  $\rm (ACF)$:
	:$$\varphi_z ({\rm \delta}t) = 2 \sigma^2 \cdot {\rm J_0}(2\pi \cdot f_{\rm D, \hspace{0.1cm} max} \cdot {\rm \delta}t)\hspace{0.05cm}.$$		:$$\varphi_z ({\rm \delta}t) = 2 \sigma^2 \cdot {\rm J_0}(2\pi \cdot f_{\rm D, \hspace{0.1cm} max} \cdot {\rm \delta}t)\hspace{0.05cm}.$$

Bene: Text replacement - "(PDS)" to "(PSD)"

2022-02-17T11:28:27Z

Text replacement - "(PDS)" to "(PSD)"

← Older revision		Revision as of 13:28, 17 February 2022
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	\hspace{0.05cm}.$$		\hspace{0.05cm}.$$

	The two channels, which are designated according to the colors "Red" and "Blue" in the graphs with  $\rm R$  and  $\rm B$  respectively, differ in the speed  $v$  and thus in the form of the power spectral density  $\rm (~~PDS~~)$   ${\it \Phi}_z(f_{\rm D})$.		The two channels, which are designated according to the colors "Red" and "Blue" in the graphs with  $\rm R$  and  $\rm B$  respectively, differ in the speed  $v$  and thus in the form of the power spectral density  $\rm (PSD)$   ${\it \Phi}_z(f_{\rm D})$.

	*In both cases, however, the PDS is a [[Mobile_Communications/Statistical_bindings_within_the_Rayleigh_process\|Jakes spectrum]].		*In both cases, however, the PDS is a [[Mobile_Communications/Statistical_bindings_within_the_Rayleigh_process\|Jakes spectrum]].

Bene: Text replacement - "power density spectrum" to "power spectral density"

2022-02-17T11:20:34Z

Text replacement - "power density spectrum" to "power spectral density"

← Older revision		Revision as of 13:20, 17 February 2022
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	\hspace{0.05cm}.$$		\hspace{0.05cm}.$$

	The two channels, which are designated according to the colors "Red" and "Blue" in the graphs with  $\rm R$  and  $\rm B$  respectively, differ in the speed  $v$  and thus in the form of the power density ~~spectrum~~  $\rm (PDS)$   ${\it \Phi}_z(f_{\rm D})$.		The two channels, which are designated according to the colors "Red" and "Blue" in the graphs with  $\rm R$  and  $\rm B$  respectively, differ in the speed  $v$  and thus in the form of the power spectral density  $\rm (PDS)$   ${\it \Phi}_z(f_{\rm D})$.

	*In both cases, however, the PDS is a [[Mobile_Communications/Statistical_bindings_within_the_Rayleigh_process\|Jakes spectrum]].		*In both cases, however, the PDS is a [[Mobile_Communications/Statistical_bindings_within_the_Rayleigh_process\|Jakes spectrum]].

Javier: Text replacement - "”" to """

2021-05-28T14:43:09Z

Text replacement - "”" to """

← Older revision		Revision as of 16:43, 28 May 2021
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	\hspace{0.05cm}.$$		\hspace{0.05cm}.$$

	The two channels, which are designated according to the colors "Red~~”~~ and "Blue~~”~~ in the graphs with  $\rm R$  and  $\rm B$  respectively, differ in the speed  $v$  and thus in the form of the power density spectrum  $\rm (PDS)$   ${\it \Phi}_z(f_{\rm D})$.		The two channels, which are designated according to the colors "Red" and "Blue" in the graphs with  $\rm R$  and  $\rm B$  respectively, differ in the speed  $v$  and thus in the form of the power density spectrum  $\rm (PDS)$   ${\it \Phi}_z(f_{\rm D})$.

	*In both cases, however, the PDS is a [[Mobile_Communications/Statistical_bindings_within_the_Rayleigh_process\|Jakes spectrum]].		*In both cases, however, the PDS is a [[Mobile_Communications/Statistical_bindings_within_the_Rayleigh_process\|Jakes spectrum]].

Javier: Text replacement - "„" to """

2021-05-28T14:35:53Z

Text replacement - "„" to """

← Older revision		Revision as of 16:35, 28 May 2021
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	\hspace{0.05cm}.$$		\hspace{0.05cm}.$$

	The two channels, which are designated according to the colors ~~&bdquo;~~Red” and ~~&bdquo;~~Blue” in the graphs with  $\rm R$  and  $\rm B$  respectively, differ in the speed  $v$  and thus in the form of the power density spectrum  $\rm (PDS)$   ${\it \Phi}_z(f_{\rm D})$.		The two channels, which are designated according to the colors "Red” and "Blue” in the graphs with  $\rm R$  and  $\rm B$  respectively, differ in the speed  $v$  and thus in the form of the power density spectrum  $\rm (PDS)$   ${\it \Phi}_z(f_{\rm D})$.

	*In both cases, however, the PDS is a [[Mobile_Communications/Statistical_bindings_within_the_Rayleigh_process\|Jakes spectrum]].		*In both cases, however, the PDS is a [[Mobile_Communications/Statistical_bindings_within_the_Rayleigh_process\|Jakes spectrum]].

Javier: Text replacement - "Category:Exercises for Mobile Communications" to "Category:Mobile Communications: Exercises"

2021-03-23T12:37:32Z

Text replacement - "Category:Exercises for Mobile Communications" to "Category:Mobile Communications: Exercises"

← Older revision		Revision as of 14:37, 23 March 2021
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	[[Category:~~Exercises for~~ Mobile Communications\|^1.3 Rayleigh Fading with Memory^]]		[[Category:Mobile Communications: Exercises\|^1.3 Rayleigh Fading with Memory^]]

Guenter at 13:31, 12 February 2021

2021-02-12T13:31:45Z

← Older revision		Revision as of 15:31, 12 February 2021
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	{{quiz-Header\|Buchseite=Mobile_Communications/Statistical_Bindings_within_the_Rayleigh_Process}}		{{quiz-Header\|Buchseite=Mobile_Communications/Statistical_Bindings_within_the_Rayleigh_Process}}

	[[File:P_ID2119__Mob_A_1_4.png\|right\|frame\| PDF and  $\|z(t)\|$  for Rayleigh ~~Fading~~ with Doppler effect]]		[[File:P_ID2119__Mob_A_1_4.png\|right\|frame\| PDF and  $\|z(t)\|$  for Rayleigh fading with Doppler effect]]
	We consider two different mobile radio channels with  [[Mobile_Communications/Wahrscheinlichkeitsdichte_des_Rayleigh%E2%80%93Fadings#Beispielhafte_Signalverl.C3.A4ufe_bei_Rayleigh.E2.80.93Fading\|Rayleigh fading]]. In both cases the PDF of the magnitude  $a(t) = \|z(t)\| ≥ 0$  is		We consider two different mobile radio channels with  [[Mobile_Communications/Wahrscheinlichkeitsdichte_des_Rayleigh%E2%80%93Fadings#Beispielhafte_Signalverl.C3.A4ufe_bei_Rayleigh.E2.80.93Fading\|Rayleigh fading]]. In both cases the PDF of the magnitude  $a(t) = \|z(t)\| ≥ 0$  is
	:$$f_a(a) = \frac{a}{\sigma^2} \cdot {\rm e}^{ -{a^2}/(2\sigma^2)}		:$$f_a(a) = \frac{a}{\sigma^2} \cdot {\rm e}^{ -{a^2}/(2\sigma^2)}