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

Hwang at 06:05, 18 September 2022

2022-09-18T06:05:01Z

2022-02-17T11:41:31Z

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

2022-02-17T11:29:04Z

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

2022-02-17T11:28:27Z

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

2022-02-17T11:20:34Z

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

2021-05-28T14:43:09Z

Text replacement - "”" to """

2021-05-28T14:35:53Z

Text replacement - "„" to """

2021-03-23T12:37:32Z

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

2021-02-12T13:31:45Z

2020-12-15T14:11:11Z

Text replacement - "===Solutions===" to "===Solution==="

@@ Line 7: / Line 7: @@
   \hspace{0.05cm}.$$
-The probability that this amount is not greater than a given value&nbsp; $A$&nbsp; is
+The probability that this magnitude is not greater than a given value&nbsp; $A$&nbsp; is
 :$${\rm Pr}(|z(t)| \le A) = 1 - {\rm e}^{ -{A^2}/(2\sigma^2)}
   \hspace{0.05cm}.$$

@@ Line 11: / Line 11: @@
   \hspace{0.05cm}.$$
-The two channels, which are designated according to the colors "Red" and "Blue" in the graphs with &nbsp;$\rm R$&nbsp; and &nbsp;$\rm B$&nbsp; respectively, differ in the speed&nbsp; $v$&nbsp; and thus in the form of the power spectral density&nbsp; $\rm (PSD)$ &nbsp; ${\it \Phi}_z(f_{\rm D})$.
+The two channels, which are designated according to the colors "Red" and "Blue" in the graphs with &nbsp;$\rm R$&nbsp; and &nbsp;$\rm B$&nbsp; respectively, differ in the speed&nbsp; $v$&nbsp; and thus in the form of the power-spectral density&nbsp; $\rm (PSD)$ &nbsp; ${\it \Phi}_z(f_{\rm D})$.
 *In both cases, however, the PDS is a [[Mobile_Communications/Statistical_bindings_within_the_Rayleigh_process|Jakes spectrum]].

← Older revision		Revision as of 11:29, 17 February 2022
Line 22:		Line 22:


−	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  $\rm (ACF)$:
	:$$\varphi_z ({\rm \delta}t) = 2 \sigma^2 \cdot {\rm J_0}(2\pi \cdot f_{\rm D, \hspace{0.1cm} max} \cdot {\rm \delta}t)\hspace{0.05cm}.$$		:$$\varphi_z ({\rm \delta}t) = 2 \sigma^2 \cdot {\rm J_0}(2\pi \cdot f_{\rm D, \hspace{0.1cm} max} \cdot {\rm \delta}t)\hspace{0.05cm}.$$

@@ Line 11: / Line 11: @@
   \hspace{0.05cm}.$$
-The two channels, which are designated according to the colors "Red" and "Blue" in the graphs with &nbsp;$\rm R$&nbsp; and &nbsp;$\rm B$&nbsp; respectively, differ in the speed&nbsp; $v$&nbsp; and thus in the form of the power spectral density&nbsp; $\rm (PDS)$ &nbsp; ${\it \Phi}_z(f_{\rm D})$.
+The two channels, which are designated according to the colors "Red" and "Blue" in the graphs with &nbsp;$\rm R$&nbsp; and &nbsp;$\rm B$&nbsp; respectively, differ in the speed&nbsp; $v$&nbsp; and thus in the form of the power spectral density&nbsp; $\rm (PSD)$ &nbsp; ${\it \Phi}_z(f_{\rm D})$.
 *In both cases, however, the PDS is a [[Mobile_Communications/Statistical_bindings_within_the_Rayleigh_process|Jakes spectrum]].

@@ Line 11: / Line 11: @@
   \hspace{0.05cm}.$$
-The two channels, which are designated according to the colors "Red" and "Blue" in the graphs with &nbsp;$\rm R$&nbsp; and &nbsp;$\rm B$&nbsp; respectively, differ in the speed&nbsp; $v$&nbsp; and thus in the form of the power density spectrum&nbsp; $\rm (PDS)$ &nbsp; ${\it \Phi}_z(f_{\rm D})$.
+The two channels, which are designated according to the colors "Red" and "Blue" in the graphs with &nbsp;$\rm R$&nbsp; and &nbsp;$\rm B$&nbsp; respectively, differ in the speed&nbsp; $v$&nbsp; and thus in the form of the power spectral density&nbsp; $\rm (PDS)$ &nbsp; ${\it \Phi}_z(f_{\rm D})$.
 *In both cases, however, the PDS is a [[Mobile_Communications/Statistical_bindings_within_the_Rayleigh_process|Jakes spectrum]].

← Older revision		Revision as of 12:37, 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^]]

@@ Line 2: / Line 2: @@
 {{quiz-Header|Buchseite=Mobile_Communications/Statistical_Bindings_within_the_Rayleigh_Process}}
-[[File:P_ID2119__Mob_A_1_4.png|right|frame| PDF and&nbsp; $|z(t)|$&nbsp; for Rayleigh Fading with Doppler effect]]
+[[File:P_ID2119__Mob_A_1_4.png|right|frame| PDF and&nbsp; $|z(t)|$&nbsp; for Rayleigh fading with Doppler effect]]
 We consider two different mobile radio channels with&nbsp; [[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&nbsp; $a(t) = |z(t)| &#8805; 0$&nbsp; is
 :$$f_a(a) = \frac{a}{\sigma^2} \cdot {\rm e}^{ -{a^2}/(2\sigma^2)}

@@ Line 76: / Line 76: @@
-===Solutions===
+===Solution===
 {{ML-Kopf}}
 '''(1)'''&nbsp; The maximum value of the PDF for both channels is&nbsp; $0.6$&nbsp; and occurs at&nbsp; $a = 1$.