Mobile Communications/Statistical Bindings within the Rayleigh Process - Revision history

Guenter at 13:31, 29 January 2023

2023-01-29T13:31:13Z

Guenter at 13:28, 29 January 2023

2023-01-29T13:28:15Z

Hwang at 13:05, 24 January 2023

2023-01-24T13:05:48Z

Hwang at 13:02, 24 January 2023

2023-01-24T13:02:37Z

Hwang at 14:06, 19 January 2023

2023-01-19T14:06:38Z

Hwang at 14:05, 19 January 2023

2023-01-19T14:05:35Z

Hwang at 13:53, 19 January 2023

2023-01-19T13:53:14Z

Hwang at 13:52, 19 January 2023

2023-01-19T13:52:06Z

Hwang at 13:50, 19 January 2023

2023-01-19T13:50:42Z

Hwang at 23:39, 12 November 2022

2022-11-12T23:39:36Z

@@ Line 58: / Line 58: @@
   \hspace{0.05cm}.</math>
-*The&nbsp; $\rm PDS$ '''KORREKTUR: PSD?''' variable is the&nbsp; [[Mobile_Communications/Statistical_Bindings_within_the_Rayleigh_Process#Doppler_frequency_and_its_distribution|$\text{Doppler frequency}$]]&nbsp; $f_{\rm D}$, because in mobile radio the so-called&nbsp; "Doppler effect"&nbsp; is the cause of the statistical dependencies. }}
+*The&nbsp; $\rm PSD$&nbsp;  variable is the&nbsp; [[Mobile_Communications/Statistical_Bindings_within_the_Rayleigh_Process#Doppler_frequency_and_its_distribution|$\text{Doppler frequency}$]]&nbsp; $f_{\rm D}$, because in mobile radio the so-called&nbsp; "Doppler effect"&nbsp; is the cause of the statistical dependencies. }}
@@ Line 183: / Line 183: @@
 == ACF and PSD with Rayleigh–Fading ==
 <br>
-We now assume an antenna with the same radiation in all directions; then the Doppler&ndash;PDS '''KORREKTUR: PSD?''' is identical in shape to the PDF of the Doppler frequencies.
+We now assume an antenna with the same radiation in all directions; then the Doppler power-spectral density  is identical in shape to the PDF of the Doppler frequencies.
-For&nbsp; ${\it \Phi}_x(f_{\rm D})$&nbsp; the PDF must still be multiplied by the power&nbsp; $\sigma^2$&nbsp; of the Gaussian process, and the resulting PDS '''KORREKTUR: PSD?'''&nbsp; ${\it \Phi}_z(f_{\rm D})$&nbsp; of the complex factor&nbsp; $z(t) = x(t) + {\rm j} \cdot y(t) $&nbsp; can be written after doubling as:
+For&nbsp; ${\it \Phi}_x(f_{\rm D})$&nbsp; the PDF must still be multiplied by the power&nbsp; $\sigma^2$&nbsp; of the Gaussian process, and the resulting PSD &nbsp; ${\it \Phi}_z(f_{\rm D})$&nbsp; of the complex factor&nbsp; $z(t) = x(t) + {\rm j} \cdot y(t) $&nbsp; can be written after doubling as:
 ::<math>{\it \Phi}_z(f_{\rm D}) =
@@ Line 208: / Line 208: @@
 The numerical values of this function can be obtained with the&nbsp; [[Applets:Bessel functions of the first kind|"applet of the same name"]]. <br>
-[[File:P_ID2117__Mob_T_1_3_S4_v2.png|right|frame|Doppler-PDS '''KORREKTUR: PSD?''' and time function (magnitude in dB) for Rayleigh fading with Doppler effect]]
+[[File:P_ID2117__Mob_T_1_3_S4_v2.png|right|frame|Doppler power-spectral density and time function (magnitude in dB) for Rayleigh fading with Doppler effect]]
 {{GraueBox|TEXT=
 $\text{Example 4:}$&nbsp; Shown on the left is the Jakes spectrum

@@ Line 7: / Line 7: @@
 == Some general remarks on ACF and PSD ==
 <br>
-The correlation between&nbsp; $r(t)$&nbsp; and&nbsp; $r(t+ \Delta t)$&nbsp; is suitable for describing the inner statistical dependencies between the neighboring signal values&nbsp;[[Theory_of_Stochastic_Signals/Auto-Correlation_Function#Auto-correlation_function_for_stationary_and_ergodic_processes|$\text{Auto-correlation function}$]]&nbsp; $\rm (ACF)$:
+The correlation between&nbsp; $r(t)$&nbsp; and&nbsp; $r(t+ \Delta t)$ &nbsp; &rArr; &nbsp; [[Theory_of_Stochastic_Signals/Auto-Correlation_Function#Auto-correlation_function_for_stationary_and_ergodic_processes|$\text{auto-correlation function}$]] &nbsp; $\rm (ACF)$ is suitable for describing the inner statistical dependencies between the neighboring signal values:
 ::<math>\varphi_r ({\rm \Delta}t) = {1}/{2} \cdot {\rm E}\big [ r(t) \cdot r^{\star}(t + {\rm \Delta}t)\big ]

@@ Line 224: / Line 224: @@
-==Exercises to the chapter==
+==Exercises for the chapter==
 <br>
 [[Aufgaben:Exercise 1.4: Rayleigh PDF and Jakes PDS]]

@@ Line 215: / Line 215: @@
-For&nbsp; [[Examples_of_Communication_Systems/Allgemeine_Beschreibung_von_GSM#Entstehung_und_Historie_von_GSM|$\text{GSM 900}$]]&nbsp; $(f_{\rm S} = 900 \ \rm MHz)$&nbsp; these values correspond to the speeds&nbsp; $v = 60 \ \rm  km/h$&nbsp; and&nbsp; $v = 120 \ \rm  km/h$,&nbsp; respectively.
+For&nbsp; [[Examples_of_Communication_Systems/General_Description_of_GSM#Emergence_and_history_of_GSM|$\text{GSM 900}$]]&nbsp; $(f_{\rm S} = 900 \ \rm MHz)$&nbsp; these values correspond to the speeds&nbsp; $v = 60 \ \rm  km/h$&nbsp; and&nbsp; $v = 120 \ \rm  km/h$,&nbsp; respectively.
-For&nbsp; [[Examples_of_Communication_Systems/Allgemeine_Beschreibung_von_GSM#Entstehung_und_Historie_von_GSM|$\text{GSM 1800}$]]&nbsp; $(f_{\rm S} = 1.8 \ \rm GHz)$&nbsp; these values apply to half as high speeds: &nbsp; $v = 30 \ \ \rm km/h$&nbsp; and&nbsp; $v = 60 \ \rm km/h$,&nbsp; resp.
+For&nbsp; [[Examples_of_Communication_Systems/General_Description_of_GSM#Emergence_and_history_of_GSM|$\text{GSM 1800}$]]&nbsp; $(f_{\rm S} = 1.8 \ \rm GHz)$&nbsp; these values apply to half as high speeds: &nbsp; $v = 30 \ \ \rm km/h$&nbsp; and&nbsp; $v = 60 \ \rm km/h$,&nbsp; resp.
 The right diagram shows the logarithmic absolute value of&nbsp; $z(t)$:

@@ Line 196: / Line 196: @@
 This process is named after&nbsp; [http://ethw.org/William_C._Jakes,_Jr. $\text{William C. Jakes Jr.}$]&nbsp; the&nbsp; &raquo;'''Jakes spectrum'''&laquo;.&nbsp; The doubling is necessary, because up to now only the contribution of the real part&nbsp; $x(t)$&nbsp; was considered. <br>
-The corresponding auto-correlation function&nbsp; $\rm (ACF)$ is obtained after a&nbsp; [[Signal_Representation/Fourier_Transform_and_Its_Inverse#The_Second_Fourier_Integral|$\text{inverse Fourier transformation}$]]:
+The corresponding auto-correlation function&nbsp; $\rm (ACF)$ is obtained after a&nbsp; [[Signal_Representation/Fourier_Transform_and_its_Inverse#The_second_Fourier_integral|$\text{inverse Fourier transformation}$]]:
 ::<math>\varphi_z ({\rm \Delta}t) =  2 \sigma^2 \cdot {\rm J_0}(2\pi \cdot f_{\rm D, \hspace{0.05cm} max} \cdot {\rm \Delta}t)\hspace{0.05cm},</math>

← Older revision		Revision as of 13:53, 19 January 2023
Line 58:		Line 58:
	\hspace{0.05cm}.</math>		\hspace{0.05cm}.</math>

−	*The  $\rm PDS$ variable is the  [[Mobile_Communications/Statistical_Bindings_within_the_Rayleigh_Process#Doppler_frequency_and_its_distribution\|$\text{Doppler frequency}$]]  $f_{\rm D}$, because in mobile radio the so-called  "Doppler effect"  is the cause of the statistical dependencies. }}	+	*The  $\rm PDS$ '''KORREKTUR: PSD?''' variable is the  [[Mobile_Communications/Statistical_Bindings_within_the_Rayleigh_Process#Doppler_frequency_and_its_distribution\|$\text{Doppler frequency}$]]  $f_{\rm D}$, because in mobile radio the so-called  "Doppler effect"  is the cause of the statistical dependencies. }}

@@ Line 7: / Line 7: @@
 == Some general remarks on ACF and PSD ==
 <br>
-The correlation between&nbsp; $r(t)$&nbsp; and&nbsp; $r(t+ \Delta t)$&nbsp; is suitable for describing the inner statistical dependencies between the neighboring signal values&nbsp;[[Theory_of_Stochastic_Signals/Auto-Correlation_Function_(ACF)#ACF_for_ergodic_processes|Autocorrelation function]]&nbsp; $\rm (ACF)$:
+The correlation between&nbsp; $r(t)$&nbsp; and&nbsp; $r(t+ \Delta t)$&nbsp; is suitable for describing the inner statistical dependencies between the neighboring signal values&nbsp;[[Theory_of_Stochastic_Signals/Auto-Correlation_Function_(ACF)#ACF_for_ergodic_processes|$\text{Autocorrelation function}$]]&nbsp; $\rm (ACF)$:
 ::<math>\varphi_r ({\rm \Delta}t) = {1}/{2} \cdot {\rm E}\big [ r(t) \cdot r^{\star}(t + {\rm \Delta}t)\big ]
@@ Line 58: / Line 58: @@
   \hspace{0.05cm}.</math>
-*The&nbsp; $\rm PDS$ variable is the&nbsp; [[Mobile_Communications/Statistical_Bindings_within_the_Rayleigh_Process#Doppler_frequency_and_its_distribution|Doppler frequency]]&nbsp; $f_{\rm D}$, because in mobile radio the so-called&nbsp; "Doppler effect"&nbsp; is the cause of the statistical dependencies. }}
+*The&nbsp; $\rm PDS$ variable is the&nbsp; [[Mobile_Communications/Statistical_Bindings_within_the_Rayleigh_Process#Doppler_frequency_and_its_distribution|$\text{Doppler frequency}$]]&nbsp; $f_{\rm D}$, because in mobile radio the so-called&nbsp; "Doppler effect"&nbsp; is the cause of the statistical dependencies. }}
@@ Line 65: / Line 65: @@
 == Phenomenological description of the Doppler effect==
 <br>
-The statistical dependencies within the real signals&nbsp; $x(t)$&nbsp; and&nbsp; $y(t)$&nbsp; or within the complex quantity&nbsp; $z(t)$&nbsp; are due to the Doppler effect.&nbsp; This was predicted theoretically in the middle of the 19th century by the Austrian mathematician, physicist and astronomer&nbsp; [https://en.wikipedia.org/wiki/Christian_Doppler Christian Andreas Doppler]&nbsp; and named after him.<br>
+The statistical dependencies within the real signals&nbsp; $x(t)$&nbsp; and&nbsp; $y(t)$&nbsp; or within the complex quantity&nbsp; $z(t)$&nbsp; are due to the Doppler effect.&nbsp; This was predicted theoretically in the middle of the 19th century by the Austrian mathematician, physicist and astronomer&nbsp; [https://en.wikipedia.org/wiki/Christian_Doppler $\text{Christian Andreas Doppler}$]&nbsp; and named after him.<br>
 {{BlaueBox|TEXT=
-$\text{Definition:}$&nbsp; The&nbsp; '''Doppler effect'''&nbsp; refers to the change in the perceived frequency of waves of any kind that occurs when the source (transmitter) and observer (receiver) move relative to each other.}}<br>
+$\text{Definition:}$&nbsp; The&nbsp; &raquo;'''Doppler effect'''&laquo;&nbsp; refers to the change in the perceived frequency of waves of any kind that occurs when the source (transmitter) and observer (receiver) move relative to each other.}}<br>
 Qualitatively, the Doppler effect can be described as follows:
@@ Line 80: / Line 80: @@
 The same effect can be seen with a&nbsp; car racing&nbsp; note.&nbsp; The frequency changes and the sound are the more obvious the faster the cars are going.}}<br>
-In this learning tutorial you can illustrate the subject matter with the interactive applet&nbsp; [[Applets:The_Doppler_Effect|The Doppler Effect]]&nbsp;.
+In this learning tutorial you can illustrate the subject matter with the interactive applet&nbsp; [[Applets:The_Doppler_Effect|"The Doppler Effect"]]&nbsp;.
 [[File:P ID2113 Mob T 1 3 S2a v1.png|right|frame|Initial situation:&nbsp; $\rm (S)$&nbsp; and&nbsp; $\rm (E)$&nbsp; are not moving|class=fit]]
@@ Line 109: / Line 109: @@
 ::<math>f_{\rm E} = f_{\rm S} \cdot \frac{\sqrt{1 - (v/c)^2} }{1 - v/c \cdot \cos(\alpha)}
 \hspace{0.05cm}.</math>
-*As the&nbsp; [[Aufgaben:Exercise_1.4Z:_On_the_Doppler_Effect|Exercise 1.4Z]]&nbsp; will show:&nbsp; One can assume at realistic speeds&nbsp; $(v \ll c)$&nbsp; the following approximation, in which the effects described by the&nbsp; [https://en.wikipedia.org/wiki/Theory_of_relativity Theory of Relativity]&nbsp; are disregarded:
+*As the&nbsp; [[Aufgaben:Exercise_1.4Z:_On_the_Doppler_Effect|"Exercise 1.4Z"]]&nbsp; will show:&nbsp; One can assume at realistic speeds&nbsp; $(v \ll c)$&nbsp; the following approximation, in which the effects described by the&nbsp; [https://en.wikipedia.org/wiki/Theory_of_relativity $\text{Theory of Relativity}$]&nbsp; are disregarded:
 ::<math>f_{\rm E} \approx f_{\rm S} \cdot \big [1 +{v}/{c} \cdot \cos(\alpha) \big ] \hspace{0.05cm}.</math>}}
@@ Line 150: / Line 150: @@
-Then for the probability of the Doppler frequency according to the chapter&nbsp; [[Theory_of_Stochastic_Signals/Exponentially_Distributed_Random_Variables#Transformation_of_random_variables|Transformation of Random Variables]]&nbsp; in the book "Stochastic Signal Theory":
+Then for the probability of the Doppler frequency according to the chapter&nbsp; [[Theory_of_Stochastic_Signals/Exponentially_Distributed_Random_Variables#Transformation_of_random_variables|"Transformation of Random Variables"]]&nbsp; in the book "Stochastic Signal Theory":
 ::<math>{\rm wdf}(f_{\rm D})=\frac{ {\rm wdf}(\alpha)}{\vert g\hspace{0.08cm}'(\alpha)\vert}\Bigg \vert_{\hspace{0.1cm} \alpha=h(f_{\rm D})}
@@ Line 194: / Line 194: @@
   \hspace{0.05cm}.</math>
-This process is named after&nbsp; [http://ethw.org/William_C._Jakes,_Jr. William C. Jakes Jr.]&nbsp; the&nbsp; '''Jakes spectrum'''.&nbsp; The doubling is necessary, because up to now only the contribution of the real part&nbsp; $x(t)$&nbsp; was considered. <br>
+This process is named after&nbsp; [http://ethw.org/William_C._Jakes,_Jr. $\text{William C. Jakes Jr.}$]&nbsp; the&nbsp; &raquo;'''Jakes spectrum'''&laquo;.&nbsp; The doubling is necessary, because up to now only the contribution of the real part&nbsp; $x(t)$&nbsp; was considered. <br>
-The corresponding auto-correlation function&nbsp; $\rm (ACF)$ is obtained after a&nbsp; [[Signal_Representation/Fourier_Transform_and_Its_Inverse#The_Second_Fourier_Integral|inverse Fourier transformation]]:
+The corresponding auto-correlation function&nbsp; $\rm (ACF)$ is obtained after a&nbsp; [[Signal_Representation/Fourier_Transform_and_Its_Inverse#The_Second_Fourier_Integral|$\text{inverse Fourier transformation}$]]:
 ::<math>\varphi_z ({\rm \Delta}t) =  2 \sigma^2 \cdot {\rm J_0}(2\pi \cdot f_{\rm D, \hspace{0.05cm} max} \cdot {\rm \Delta}t)\hspace{0.05cm},</math>
@@ Line 206: / Line 206: @@
   \hspace{0.05cm}.</math>
-The numerical values of this function can be obtained with the&nbsp; [[Applets:Bessel functions of the first kind|applet of the same name]]. <br>
+The numerical values of this function can be obtained with the&nbsp; [[Applets:Bessel functions of the first kind|"applet of the same name"]]. <br>
 [[File:P_ID2117__Mob_T_1_3_S4_v2.png|right|frame|Doppler-PDS and time function (magnitude in dB) for Rayleigh fading with Doppler effect]]

@@ Line 61: / Line 61: @@
-This effect is explained on the next page.
+This effect is explained in the next section.
 == Phenomenological description of the Doppler effect==
@@ Line 114: / Line 114: @@
 == Doppler frequency and its distribution==
 <br>
-We summarize the statements of the last page briefly, whereby we start from the second, i.e. the non&ndash;relativistic equation:
+We summarize the statements of the last section briefly, whereby we start from the second, i.e. the non&ndash;relativistic equation:
 *A relative movement between transmitter (source) and receiver (observer) results in a shift by the Doppler frequency&nbsp; $f_{\rm D} = f_{\rm E} - f_{\rm S}$.