Mobile Communications/Probability Density of Rayleigh Fading - Revision history

Guenter at 13:40, 1 February 2023

2023-02-01T13:40:39Z

Hwang at 13:06, 24 January 2023

2023-01-24T13:06:40Z

Hwang at 14:25, 23 January 2023

2023-01-23T14:25:22Z

Hwang at 13:47, 19 January 2023

2023-01-19T13:47:19Z

Hwang at 13:29, 19 January 2023

2023-01-19T13:29:40Z

Hwang at 09:56, 17 November 2022

2022-11-17T09:56:47Z

Guenter at 10:45, 13 February 2021

2021-02-13T10:45:45Z

Guenter at 11:31, 12 February 2021

2021-02-12T11:31:45Z

Javier: Text replacement - "low pass" to "low-pass"

2020-12-15T14:19:32Z

Text replacement - "low pass" to "low-pass"

Guenter at 15:34, 7 December 2020

2020-12-07T15:34:36Z

@@ Line 78: / Line 78: @@
 One speaks in this case of&nbsp; &raquo;'''non-frequency selective fading'''&laquo; &nbsp; or&nbsp; &raquo;'''Flat Fading'''&laquo;&nbsp; or&nbsp; &raquo;'''Rayleigh Fading'''&laquo;. }}
-== Modeling of non-frequency selective fading==
+== Modeling of non-frequency-selective fading==
 <br>
 The figure shows the model for generating non-frequency selective fading &nbsp; &rArr; &nbsp; Rayleigh fading.<br>

@@ Line 183: / Line 183: @@
 [[Aufgaben:Exercise 1.3: Rayleigh Fading|"Exercise 1.3"]]&nbsp; and the&nbsp; [[Aufgaben:Exercise 1.3Z: Rayleigh Fading Revisited|"Exercise 1.3Z"]].<br>
-==Exercises to the chapter==
+==Exercises for the chapter==
 <br>
 [[Aufgaben:Exercise 1.3: Rayleigh Fading]]

@@ Line 82: / Line 82: @@
 The figure shows the model for generating non-frequency selective fading &nbsp; &rArr; &nbsp; Rayleigh fading.<br>
-*The received signal&nbsp; $r(t)$&nbsp; is obtained by multiplying the transmission signal&nbsp; $s(t)$&nbsp; by the time function&nbsp; $z(t)$.
+*The received signal&nbsp; $r(t)$&nbsp; is obtained by multiplying the transmitted signal&nbsp; $s(t)$&nbsp; by the time function&nbsp; $z(t)$.
 *It should be remembered again that all signals or time functions&nbsp; $s(t)$,&nbsp; $z(t)$&nbsp; and&nbsp; $r(t)$&nbsp; refer to the equivalent low-pass range.

@@ Line 155: / Line 155: @@
 From these descriptions it is clear:
-*The magnitude&nbsp; $a(t) =\vert z(t)\vert$&nbsp; has a&nbsp; [[Theory_of_Stochastic_Signals/Weitere_Verteilungen#Rayleighverteilung|$\text{Rayleigh PDF}$]] &nbsp;&#8658;&nbsp; hence the name "Rayleigh fading":
+*The magnitude&nbsp; $a(t) =\vert z(t)\vert$&nbsp; has a&nbsp; [[Theory_of_Stochastic_Signals/Further_Distributions#Rayleigh_PDF|$\text{Rayleigh PDF}$]] &nbsp;&#8658;&nbsp; hence the name "Rayleigh fading":
 ::<math>f_a(a) =

@@ Line 38: / Line 38: @@
 {{BlaueBox|TEXT=
-$\text{Conclusion:}$&nbsp; This gives the following intermediate result for mobile radio: &nbsp; The&nbsp; '''received signal in the equivalent low-pass range''' &nbsp; can be represented as
+$\text{Conclusion:}$&nbsp; This gives the following intermediate result for mobile radio: &nbsp; The&nbsp; &raquo;'''received signal in the equivalent low-pass range'''&laquo; &nbsp; can be represented as
 ::<math>r(t)=  \sum_{m=1}^{M} z_m(t) \cdot  s(t - \tau_{m}) \hspace{0.5cm} {\rm with}
@@ Line 57: / Line 57: @@
 two important special cases can be derived:
-*If there is more than one main path&nbsp; $(M \ge 2)$, one speaks of&nbsp; <b>multipath propagation</b>. &nbsp; As will be shown in the second main chapter &nbsp; &rArr; &nbsp; [[Mobile_Communications/General_description_of_time_variant_systems|Frequency-selective transmission channels]]&nbsp; then&nbsp; &ndash;&nbsp; depending on the frequency&nbsp; &ndash;&nbsp; constructive or destructive overlaps up to complete extinction occur.
+*If there is more than one main path&nbsp; $(M \ge 2)$, one speaks of&nbsp; <b>multipath propagation</b>. &nbsp; As will be shown in the second main chapter &nbsp; &rArr; &nbsp; [[Mobile_Communications/General_description_of_time_variant_systems|"Frequency-selective transmission channels"]]&nbsp; then&nbsp; &ndash;&nbsp; depending on the frequency&nbsp; &ndash;&nbsp; constructive or destructive overlaps up to complete extinction occur.
-*For some frequencies, multipath propagation proves to be favourable, for others, very unfavourable.&nbsp; This effect is called&nbsp; '''frequency selective fading'''.<br>
+*For some frequencies, multipath propagation proves to be favourable, for others, very unfavourable.&nbsp; This effect is called&nbsp; &raquo;'''frequency selective fading'''&laquo;.<br>
@@ Line 76: / Line 76: @@
   \phi_{n}(t)}
   \hspace{0.05cm}. </math>
-One speaks in this case of&nbsp; '''non-frequency selective fading''' &nbsp; or&nbsp; '''Flat Fading'''&nbsp; or&nbsp; '''Rayleigh Fading'''. }}
+One speaks in this case of&nbsp; &raquo;'''non-frequency selective fading'''&laquo; &nbsp; or&nbsp; &raquo;'''Flat Fading'''&laquo;&nbsp; or&nbsp; &raquo;'''Rayleigh Fading'''&laquo;. }}
 == Modeling of non-frequency selective fading==
@@ Line 101: / Line 101: @@
 It should be noted about this equation and the above graph:
 *The time dependent attenuation&nbsp; $\alpha_{n}(t)$&nbsp; and the time dependent phase&nbsp; $\phi_{n}(t)$&nbsp; depend on the environmental conditions.
-* $\phi_{n}(t)$&nbsp; captures the slightly different delays on the&nbsp; $N$&nbsp; paths and the&nbsp; [[Mobile_Communications/Statistical_Properties_within_the_Rayleigh_Process#Doppler_frequency_and_its_distribution|Doppler effect]]&nbsp; due to the movement.
+* $\phi_{n}(t)$&nbsp; captures the slightly different delays on the&nbsp; $N$&nbsp; paths and the&nbsp; [[Mobile_Communications/Statistical_Properties_within_the_Rayleigh_Process#Doppler_frequency_and_its_distribution|$\text{Doppler effect}$]]&nbsp; due to the movement.
 *The time function&nbsp; $z(t)$&nbsp; is a complex quantity whose real and imaginary part denoted in the following as&nbsp; $x(t)$&nbsp; and&nbsp; $y(t)$&nbsp;.
 *A deterministic description of the random variable&nbsp; $z(t) = x(t) + {\rm j}\cdot y(t)$&nbsp; is not possible;&nbsp; the functions&nbsp; $x(t)$&nbsp; and&nbsp; $y(t)$ must be modeled by stochastic processes.
-*If the number&nbsp; $N$&nbsp; of the&nbsp; (slightly)&nbsp; different delays is sufficiently large, then according to the&nbsp; [https://en.wikipedia.org/wiki/Central_limit_theorem Central Limit Theorem]&nbsp; for this&nbsp; [[Theory_of_Stochastic_Signals/Gaußverteilte_Zufallsgrößen|Gaussian Random Variable]].
+*If the number&nbsp; $N$&nbsp; of the&nbsp; (slightly)&nbsp; different delays is sufficiently large, then according to the&nbsp; [https://en.wikipedia.org/wiki/Central_limit_theorem $\text{Central Limit Theorem}$]&nbsp; for this&nbsp; [[Theory_of_Stochastic_Signals/Gaußverteilte_Zufallsgrößen|$\text{Gaussian Random Variable}$]].
 *The two components&nbsp; $x(t)$&nbsp; and&nbsp; $y(t)$&nbsp; are each mean-free and have the same variance&nbsp; $\sigma^2$:
@@ Line 115: / Line 115: @@
 *We observe the orthogonality of the real part and the imaginary part&nbsp; (both cosine and sine of the same argument).&nbsp; Thus the two components are also uncorrelated.&nbsp; Only in the case of Gaussian random variables does the statistical independence of&nbsp; $x(t)$&nbsp; and&nbsp; $y(t)$ follow from this.
-*Because of the Doppler effect, however, there are statistical dependencies within the real part&nbsp; $x(t)$&nbsp; and within the imaginary part&nbsp; $y(t)$.&nbsp; These two quantities are created in the above model by two&nbsp; [[Theory_of_Stochastic_Signals/Digital_Filters|Digital Filters]].
+*Because of the Doppler effect, however, there are statistical dependencies within the real part&nbsp; $x(t)$&nbsp; and within the imaginary part&nbsp; $y(t)$.&nbsp; These two quantities are created in the above model by two&nbsp; [[Theory_of_Stochastic_Signals/Digital_Filters|$\text{Digital Filters}$]].
 == Exemplary signal curves with Rayleigh fading==
@@ Line 128: / Line 128: @@
 *The real part is Gaussian distributed&nbsp; (see upper right graph), as shown in the signal&nbsp; $x(t)$.&nbsp; &nbsp; Red is the Gaussian PDF $f_x(x)$ and blue is the histogram obtained by simulation over&nbsp; $10\hspace{0.05cm}000$&nbsp; samples.
-*The parameter used was a&nbsp; [[Mobile_Communications/Statistical_Bindings_within_the_Rayleigh_Process#Doppler frequency and its distribution|maximum Doppler frequency]]&nbsp; of&nbsp; $f_{\rm D, \ max} = 100 \ \rm Hz$.&nbsp; Therefore there are statistical bindings within the functions&nbsp; $x(t)$&nbsp; and&nbsp; $y(t)$.&nbsp; More details about the Doppler effect can be found in the next chapter.
+*The parameter used was a&nbsp; [[Mobile_Communications/Statistical_Bindings_within_the_Rayleigh_Process#Doppler frequency and its distribution|$\text{maximum Doppler frequency}$]]&nbsp; of&nbsp; $f_{\rm D, \ max} = 100 \ \rm Hz$.&nbsp; Therefore there are statistical bindings within the functions&nbsp; $x(t)$&nbsp; and&nbsp; $y(t)$.&nbsp; More details about the Doppler effect can be found in the next chapter.
 *The PDF&nbsp; $f_y(y)$&nbsp; of the imaginary part is identical to&nbsp; $f_x(x)$.&nbsp; The variance is&nbsp; $\sigma_x^2 =\sigma_y^2 = 0. 5 \ (=\sigma^2)$.&nbsp; Between&nbsp; $x(t)$&nbsp; and&nbsp; $y(t)$&nbsp; there are no statistical bindings;&nbsp; the signals are orthogonal.
@@ Line 155: / Line 155: @@
 From these descriptions it is clear:
-*The magnitude&nbsp; $a(t) =\vert z(t)\vert$&nbsp; has a&nbsp; [[Theory_of_Stochastic_Signals/Weitere_Verteilungen#Rayleighverteilung| Rayleigh PDF]] &nbsp;&#8658;&nbsp; hence the name "Rayleigh fading":
+*The magnitude&nbsp; $a(t) =\vert z(t)\vert$&nbsp; has a&nbsp; [[Theory_of_Stochastic_Signals/Weitere_Verteilungen#Rayleighverteilung|$\text{Rayleigh PDF}$]] &nbsp;&#8658;&nbsp; hence the name "Rayleigh fading":
 ::<math>f_a(a) =
@@ Line 170: / Line 170: @@
   \hspace{0.05cm}.  </math>
-*The PDF of the absolute value square&nbsp; $p(t)$&nbsp; is given by&nbsp; [[Theory_of_Stochastic_Signals/Exponentially_Distributed_Random_Variables#Transformation_of_random_variables| nonlinear transformation]]&nbsp; the PDF $f_a(a)$ &nbsp; &rArr; &nbsp; $f_p(p)$&nbsp; is exponentially distributed:
+*The PDF of the absolute value square&nbsp; $p(t)$&nbsp; is given by&nbsp; [[Theory_of_Stochastic_Signals/Exponentially_Distributed_Random_Variables#Transformation_of_random_variables|$\text{nonlinear transformation}$]]&nbsp; the PDF $f_a(a)$ &nbsp; &rArr; &nbsp; $f_p(p)$&nbsp; is exponentially distributed:
 ::<math>f_p(p) \hspace{-0.05cm}=\hspace{-0.05cm}
@@ Line 181: / Line 181: @@
 Further information about the&nbsp; Rayleigh fading&nbsp; can be found in the&nbsp;
-[[Aufgaben:Exercise 1.3: Rayleigh Fading|Exercise 1.3]]&nbsp; and the&nbsp; [[Aufgaben:Exercise 1.3Z: Rayleigh Fading Revisited|Exercise 1.3Z]].<br>
+[[Aufgaben:Exercise 1.3: Rayleigh Fading|"Exercise 1.3"]]&nbsp; and the&nbsp; [[Aufgaben:Exercise 1.3Z: Rayleigh Fading Revisited|"Exercise 1.3Z"]].<br>
 ==Exercises to the chapter==

@@ Line 87: / Line 87: @@
 [[File:EN_Mob_T_1_2_S2.png|center|frame|Rayleigh fading channel model|class=fit]]
-We now look at the multiplicative error&nbsp; $z(t)\ne 1$&nbsp; according to this Rayleigh model more precisely.&nbsp; For the complex coefficient applies according to the last page:
+We now look at the multiplicative error&nbsp; $z(t)\ne 1$&nbsp; according to this Rayleigh model more precisely.&nbsp; For the complex coefficient applies according to the last section:
 ::<math>z(t) = \sum_{n=1}^{N}  \alpha_{n}(t) \cdot

@@ Line 32: / Line 32: @@
   \hspace{0.05cm}.</math>
-The two equations given so far are identical.&nbsp; A simplification results only if one replaces for each main path&nbsp; $m \in \{1, \hspace{0.04cm}\text{...}\hspace{0.04cm}, M\}$&nbsp; the&nbsp; $N_m$&nbsp; delays, which differ slightly due to reflections at fine structures as well as possibly due to diffraction&ndash; and refraction phenomena, by a mean delay:
+The two equations given so far are identical.&nbsp; A simplification results only if one replaces for each main path&nbsp; $m \in \{1, \hspace{0.04cm}\text{...}\hspace{0.04cm}, M\}$&nbsp; the&nbsp; $N_m$&nbsp; delays, which differ slightly due to reflections at fine structures as well as possibly due to diffraction and refraction phenomena, by a mean delay:
 ::<math>\tau_{m} =  \frac{1}{N_m} \cdot  \sum_{n=1}^{N_m} \tau_{m,\hspace{0.04cm}n}

@@ Line 7: / Line 7: @@
 == A very general description of the mobile communication channel ==
 <br>
-To simplify the notation, the addition "TP" (Tiefpass &nbsp; &rArr; &nbsp; low pass)&nbsp; is omitted in the following.&nbsp; Thus the real signal&nbsp; $s(t) = 1$&nbsp; is present at the input of the mobile radio channel and the output signal&nbsp; $r(t)$&nbsp; is complex-valued.&nbsp; Additional noise processes are excluded.<br>
+To simplify the notation, the addition "TP" (Tiefpass &nbsp; &rArr; &nbsp; low-pass)&nbsp; is omitted in the following.&nbsp; Thus the real signal&nbsp; $s(t) = 1$&nbsp; is present at the input of the mobile radio channel and the output signal&nbsp; $r(t)$&nbsp; is complex-valued.&nbsp; Additional noise processes are excluded.<br>
-The radio signal&nbsp; $s(t)$&nbsp; can reach the receiver via a large number of paths, whereby the individual signal components are attenuated in different ways and delayed for different lengths.&nbsp; In general, it is possible to express the received signal&nbsp; (in the equivalent low pass range)&nbsp; without taking thermal noise into account as it follows:
+The radio signal&nbsp; $s(t)$&nbsp; can reach the receiver via a large number of paths, whereby the individual signal components are attenuated in different ways and delayed for different lengths.&nbsp; In general, it is possible to express the received signal&nbsp; (in the equivalent low-pass range)&nbsp; without taking thermal noise into account as it follows:
 ::<math>r(t)=  \sum_{k=1}^{K}  \alpha_{k}(t) \cdot {\rm e}^{\hspace{0.05cm}{\rm j}\hspace{0.02cm}\cdot \hspace{0.02cm} \phi_{k}(t)} \cdot s(t - \tau_{k})
@@ Line 38: / Line 38: @@
 {{BlaueBox|TEXT=
-$\text{Conclusion:}$&nbsp; This gives the following intermediate result for mobile radio: &nbsp; The&nbsp; ''received signal in the equivalent low pass range'' &nbsp; can be represented as
+$\text{Conclusion:}$&nbsp; This gives the following intermediate result for mobile radio: &nbsp; The&nbsp; ''received signal in the equivalent low-pass range'' &nbsp; can be represented as
 ::<math>r(t)=  \sum_{m=1}^{M} z_m(t) \cdot  s(t - \tau_{m}) \hspace{0.5cm} {\rm with}
@@ Line 87: / Line 87: @@
 *The received signal&nbsp; $r(t)$&nbsp; is obtained by multiplying the transmission signal&nbsp; $s(t)$&nbsp; by the time function&nbsp; $z(t)$&nbsp;.
-*It should be remembered again that all signals or time functions&nbsp; $s(t)$,&nbsp; $z(t)$&nbsp; and&nbsp; $r(t)$&nbsp; refer to the equivalent low pass range.
+*It should be remembered again that all signals or time functions&nbsp; $s(t)$,&nbsp; $z(t)$&nbsp; and&nbsp; $r(t)$&nbsp; refer to the equivalent low-pass range.
 <br clear=all>We now look at the multiplicative error&nbsp; $z(t)\ne 1$&nbsp; according to this Rayleigh model more precisely.&nbsp; For the complex coefficient applies according to the last page:

@@ Line 78: / Line 78: @@
 One speaks in this case
 *of&nbsp; '''non-frequency selective fading''' &nbsp;
-*or&nbsp; '''Flat&ndash;Fading</i>&nbsp;
+*or&nbsp; '''Flat Fading</i>&nbsp;
-*or&nbsp; '''Rayleigh&ndash;Fading'''. }}
+*or&nbsp; '''Rayleigh Fading'''. }}
 == Modeling of non-frequency selective fading==