Mobile Communications/Non-Frequency-Selective Fading With Direct Component - Revision history

Guenter: Guenter moved page Mobile Communications/Non-Frequency Selective Fading With Direct Component to Mobile Communications/Non-Frequency-Selective Fading With Direct Component

2023-02-03T14:55:42Z

Guenter moved page Mobile Communications/Non-Frequency Selective Fading With Direct Component to Mobile Communications/Non-Frequency-Selective Fading With Direct Component

Hwang at 13:17, 24 January 2023

2023-01-24T13:17:38Z

Hwang at 13:13, 24 January 2023

2023-01-24T13:13:23Z

Hwang at 13:10, 24 January 2023

2023-01-24T13:10:38Z

Hwang at 13:09, 24 January 2023

2023-01-24T13:09:21Z

Hwang at 14:26, 23 January 2023

2023-01-23T14:26:56Z

Bene: Text replacement - "List of sources" to "References"

2022-02-17T11:39:17Z

Text replacement - "List of sources" to "References"

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

2022-02-17T11:28:27Z

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

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

2022-02-15T15:32:35Z

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

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

2022-02-15T15:30:42Z

Text replacement - " PDS " to " PSD "

@@ Line 64: / Line 64: @@
 *The PDF $f_\phi(\phi)$&nbsp; shows the preferred angle&nbsp; $\phi \approx -45^\circ$&nbsp; of the given Rice channel &nbsp; The complex factor&nbsp; $z(t)$&nbsp; is located mainly in the fourth quadrant because of&nbsp; $x_0 > 0$&nbsp; and&nbsp; $y_0 < 0$&nbsp;, whereas in the Rayleigh channel all quadrants are equally probable.<br>
-==Exercises to the chapter==
+==Exercises for the chapter==
 <br>
 [[Aufgaben:Exercise 1.6: Autocorrelation Function and PSD with Rice Fading]]

@@ Line 19: / Line 19: @@
 <br clear=all>
 The Rice fading model can be summarized as follows, see also&nbsp; [Hin08]<ref name = 'Hin08'>Hindelang, T.:&nbsp; Mobile Communications. &nbsp; Lecture notes. Institute for Communications Engineering. &nbsp; Technical University of Munich, 2008.</ref>:
-*The real part&nbsp; $x(t)$&nbsp; is gaussian distributed with mean value&nbsp; $x_0$&nbsp; and variance&nbsp; $\sigma ^2$.
+*The real part&nbsp; $x(t)$&nbsp; is Gaussian distributed with mean value&nbsp; $x_0$&nbsp; and variance&nbsp; $\sigma ^2$.
-*The imaginary part&nbsp; $y(t)$&nbsp; is also gaussian distributed&nbsp; $($mean&nbsp; $y_0$,&nbsp; equal variance&nbsp; $\sigma ^2)$&nbsp; and independent of&nbsp; $x(t)$.<br>
+*The imaginary part&nbsp; $y(t)$&nbsp; is also Gaussian distributed&nbsp; $($mean&nbsp; $y_0$,&nbsp; equal variance&nbsp; $\sigma ^2)$&nbsp; and independent of&nbsp; $x(t)$.<br>
 *For&nbsp; $z_0 \ne 0$&nbsp; the value&nbsp; $|z(t)|$&nbsp; has a [[Theory_of_Stochastic_Signals/Further_distributions#Rice_PDF|$\text{Rice PDF}$]], from which the term&nbsp; "Rice fading"&nbsp; is derived.

@@ Line 7: / Line 7: @@
 == Channel model and Rice PDF ==
 <br>
-The&nbsp; [[Mobile_Communications/Probability_Density_of_Rayleigh_Fading#Modeling_of_non-frequency_selective_fading| \text{Rayleigh distribution}$]]&nbsp; describes the mobile communication channel under the assumption that there is no direct path and thus the multiplicative factor&nbsp; $z(t) = x(t) + {\rm j} \cdot y(t)$&nbsp; is solely composed of diffusely scattered components.
+The&nbsp; [[Mobile_Communications/Probability_Density_of_Rayleigh_Fading#Modeling_of_non-frequency_selective_fading| $\text{Rayleigh distribution}$]]&nbsp; describes the mobile communication channel under the assumption that there is no direct path and thus the multiplicative factor&nbsp; $z(t) = x(t) + {\rm j} \cdot y(t)$&nbsp; is solely composed of diffusely scattered components.
 If a direct component&nbsp; $($Line of Sight,&nbsp; $\rm LoS)$&nbsp; is present, it is necessary to add direct components &nbsp; $x_0$&nbsp; and/or&nbsp; $y_0$&nbsp; to the zero mean Gaussian processes &nbsp; $x(t)$&nbsp; and&nbsp; $y(t)$:
@@ Line 22: / Line 22: @@
 *The imaginary part&nbsp; $y(t)$&nbsp; is also gaussian distributed&nbsp; $($mean&nbsp; $y_0$,&nbsp; equal variance&nbsp; $\sigma ^2)$&nbsp; and independent of&nbsp; $x(t)$.<br>
-*For&nbsp; $z_0 \ne 0$&nbsp; the value&nbsp; $|z(t)|$&nbsp; has a [[Theory_of_Stochastic_Signals/Further_distributions#Rice_PDF|\text{Rice PDF}$]], from which the term&nbsp; "Rice fading"&nbsp; is derived.
+*For&nbsp; $z_0 \ne 0$&nbsp; the value&nbsp; $|z(t)|$&nbsp; has a [[Theory_of_Stochastic_Signals/Further_distributions#Rice_PDF|$\text{Rice PDF}$]], from which the term&nbsp; "Rice fading"&nbsp; is derived.
 *To simplify the notation we set&nbsp; $|z(t)| = a(t)$. &nbsp; For&nbsp; $a < 0$&nbsp; it's PDF is&nbsp; $f_a(a) \equiv 0$,&nbsp; for&nbsp; $a \ge 0$ the following equation applies, where&nbsp; $\rm I_0(\cdot)$&nbsp; denotes the&nbsp; "modified Bessel&ndash;function" of zero order:
@@ Line 45: / Line 45: @@
 :$$x_0 = 0.707,\ \ y_0 = -0.707.$$
-For the generation of the signals according to the above model, the&nbsp; [[Mobile_Communications/Statistical_Bindings_within_the_Rayleigh_Process#Doppler_frequency_and_its_distribution|\text{maximum Doppler frequency}$]]&nbsp; $f_\text{D, max} = 100 \ \rm Hz$&nbsp; was used as reference.
+For the generation of the signals according to the above model, the&nbsp; [[Mobile_Communications/Statistical_Bindings_within_the_Rayleigh_Process#Doppler_frequency_and_its_distribution|$\text{maximum Doppler frequency}$]]&nbsp; $f_\text{D, max} = 100 \ \rm Hz$&nbsp; was used as reference.
 The auto-correlation function&nbsp; $\rm (ACF)$&nbsp; and power-spectral density&nbsp; $\rm (PSD)$&nbsp; of Rayleigh and Rice differ only slightly, other than adjusted parameter values.&nbsp; The following applies:

@@ Line 7: / Line 7: @@
 == Channel model and Rice PDF ==
 <br>
-The&nbsp; [[Mobile_Communications/Probability_Density_of_Rayleigh_Fading#Modeling_of_non-frequency_selective_fading| Rayleigh distribution]]&nbsp; describes the mobile communication channel under the assumption that there is no direct path and thus the multiplicative factor&nbsp; $z(t) = x(t) + {\rm j} \cdot y(t)$&nbsp; is solely composed of diffusely scattered components.
+The&nbsp; [[Mobile_Communications/Probability_Density_of_Rayleigh_Fading#Modeling_of_non-frequency_selective_fading| \text{Rayleigh distribution}$]]&nbsp; describes the mobile communication channel under the assumption that there is no direct path and thus the multiplicative factor&nbsp; $z(t) = x(t) + {\rm j} \cdot y(t)$&nbsp; is solely composed of diffusely scattered components.
 If a direct component&nbsp; $($Line of Sight,&nbsp; $\rm LoS)$&nbsp; is present, it is necessary to add direct components &nbsp; $x_0$&nbsp; and/or&nbsp; $y_0$&nbsp; to the zero mean Gaussian processes &nbsp; $x(t)$&nbsp; and&nbsp; $y(t)$:
@@ Line 16: / Line 16: @@
 ::<math>z(t) = x(t) + {\rm j} \cdot y(t) \hspace{0.1cm} \Rightarrow \hspace{0.1cm} z(t) +z_0 \hspace{0.05cm},\hspace{0.2cm}
   z_0 = x_0 + {\rm j} \cdot y_0\hspace{0.05cm}.</math>
-The graph shows this&nbsp; '''Rice fading channel model'''.&nbsp; As a special case, the Rayleigh model results when &nbsp; $x_0 = y_0= 0$.
+The graph shows this&nbsp; &raquo;'''Rice fading channel model'''&laquo;.&nbsp; As a special case, the Rayleigh model results when &nbsp; $x_0 = y_0= 0$.
 <br clear=all>
 The Rice fading model can be summarized as follows, see also&nbsp; [Hin08]<ref name = 'Hin08'>Hindelang, T.:&nbsp; Mobile Communications. &nbsp; Lecture notes. Institute for Communications Engineering. &nbsp; Technical University of Munich, 2008.</ref>:
@@ Line 22: / Line 22: @@
 *The imaginary part&nbsp; $y(t)$&nbsp; is also gaussian distributed&nbsp; $($mean&nbsp; $y_0$,&nbsp; equal variance&nbsp; $\sigma ^2)$&nbsp; and independent of&nbsp; $x(t)$.<br>
-*For&nbsp; $z_0 \ne 0$&nbsp; the value&nbsp; $|z(t)|$&nbsp; has a [[Theory_of_Stochastic_Signals/Further_distributions#Rice_PDF| Rice PDF]], from which the term&nbsp; "Rice fading"&nbsp; is derived.
+*For&nbsp; $z_0 \ne 0$&nbsp; the value&nbsp; $|z(t)|$&nbsp; has a [[Theory_of_Stochastic_Signals/Further_distributions#Rice_PDF|\text{Rice PDF}$]], from which the term&nbsp; "Rice fading"&nbsp; is derived.
 *To simplify the notation we set&nbsp; $|z(t)| = a(t)$. &nbsp; For&nbsp; $a < 0$&nbsp; it's PDF is&nbsp; $f_a(a) \equiv 0$,&nbsp; for&nbsp; $a \ge 0$ the following equation applies, where&nbsp; $\rm I_0(\cdot)$&nbsp; denotes the&nbsp; "modified Bessel&ndash;function" of zero order:
-::<math>f_a(a) = \frac{a}{\sigma^2} \cdot {\rm exp} \big [ -\frac{a^2 + |z_0|^2}{2\sigma^2}\big ] \cdot {\rm I}_0 \left [ \frac{a \cdot |z_0|}{\sigma^2} \right ]\hspace{0.5cm}\text{mit}\hspace{0.5cm}{\rm I }_0 (u) = {\rm J }_0 ({\rm j} \cdot u) =
+::<math>f_a(a) = \frac{a}{\sigma^2} \cdot {\rm exp} \big [ -\frac{a^2 + |z_0|^2}{2\sigma^2}\big ] \cdot {\rm I}_0 \left [ \frac{a \cdot |z_0|}{\sigma^2} \right ]\hspace{0.5cm}\text{with}\hspace{0.5cm}{\rm I }_0 (u) = {\rm J }_0 ({\rm j} \cdot u) =
   \sum_{k = 0}^{\infty} \frac{ (u/2)^{2k}}{k! \cdot \Gamma (k+1)}
   \hspace{0.05cm}.</math>
@@ Line 45: / Line 45: @@
 :$$x_0 = 0.707,\ \ y_0 = -0.707.$$
-For the generation of the signals according to the above model, the&nbsp; [[Mobile_Communications/Statistical_Bindings_within_the_Rayleigh_Process#Doppler_frequency_and_its_distribution|maximum Doppler frequency]]&nbsp; $f_\text{D, max} = 100 \ \rm Hz$&nbsp; was used as reference.
+For the generation of the signals according to the above model, the&nbsp; [[Mobile_Communications/Statistical_Bindings_within_the_Rayleigh_Process#Doppler_frequency_and_its_distribution|\text{maximum Doppler frequency}$]]&nbsp; $f_\text{D, max} = 100 \ \rm Hz$&nbsp; was used as reference.
 The auto-correlation function&nbsp; $\rm (ACF)$&nbsp; and power-spectral density&nbsp; $\rm (PSD)$&nbsp; of Rayleigh and Rice differ only slightly, other than adjusted parameter values.&nbsp; The following applies:

@@ Line 52: / Line 52: @@
 ::<math> {\it \Phi}_z(f_{\rm D})\Bigg |_{\hspace{0.1cm}{\rm Rice}} \hspace{-0.5cm}  =    {\it \Phi}_z(f_{\rm D})\Bigg |_{\hspace{0.1cm}{\rm Rayleigh}} \hspace{-0.8cm} + |z_0|^2 \cdot \delta (f_{\rm D}) \hspace{0.05cm}.</math>
-It is taken into account that the spectral representation of a DC component leads to a Dirac function.<br>
+It is taken into account that the spectral representation of a DC component leads to a Dirac delta function.<br>
 <br clear= all>
 It should be noted about this graph:

← Older revision	Revision as of 14:55, 3 February 2023
(No difference)

@@ Line 72: / Line 72: @@
 [[Aufgaben:Exercise 1.7: PDF of Rice Fading]]
-==List of sources==
+==References==
 {{Display}}

@@ Line 47: / Line 47: @@
 For the generation of the signals according to the above model, the&nbsp; [[Mobile_Communications/Statistical_Bindings_within_the_Rayleigh_Process#Doppler_frequency_and_its_distribution|maximum Doppler frequency]]&nbsp; $f_\text{D, max} = 100 \ \rm Hz$&nbsp; was used as reference.
-The auto-correlation function&nbsp; $\rm (ACF)$&nbsp; and power-spectral density&nbsp; $\rm (PDS)$&nbsp; of Rayleigh and Rice differ only slightly, other than adjusted parameter values.&nbsp; The following applies:
+The auto-correlation function&nbsp; $\rm (ACF)$&nbsp; and power-spectral density&nbsp; $\rm (PSD)$&nbsp; of Rayleigh and Rice differ only slightly, other than adjusted parameter values.&nbsp; The following applies:
 ::<math>\varphi_z ({\rm \Delta}t)\Bigg |_{\hspace{0.1cm}{\rm Rice}}  \hspace{-0.5cm}  =  \varphi_z ({\rm \Delta}t)\Bigg |_{\hspace{0.1cm}{\rm Rayleigh}} \hspace{-0.8cm} + |z_0|^2 \hspace{0.05cm},</math>

@@ Line 66: / Line 66: @@
 ==Exercises to the chapter==
 <br>
-[[Aufgaben:Exercise 1.6: Autocorrelation Function and PDS with Rice Fading]]
+[[Aufgaben:Exercise 1.6: Autocorrelation Function and PSD with Rice Fading]]
 [[Aufgaben:Exercise 1.6Z: Comparison of Rayleigh and Rice]]