Mobile Communications/The GWSSUS Channel Model - Revision history

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2026-03-16T13:47:04Z

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	{{Display}}		{{Display}}
			[[de:Mobile_Kommunikation/Das_GWSSUS–Kanalmodell]]

Guenter at 16:38, 30 January 2023

2023-01-30T16:38:16Z

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	The general relationship between the four system functions is very complicated due to non-stationary effects.		The general relationship between the four system functions is very complicated due to non-stationary effects.

	[[File:EN_Mob_T_2_3_S2_v2.png\|right\|frame\|Connections between the description functions of the GWSSUS model ~~'''KORREKTUR: PSD'''~~\|class=fit]]		[[File:EN_Mob_T_2_3_S2_v2.png\|right\|frame\|Connections between the description functions of the GWSSUS model\|class=fit]]
	Compared to the general model, some limitations have to be made in order to arrive at a suitable model for the mobile communications channel from which relevant statements for practical applications can be derived.<br>		Compared to the general model, some limitations have to be made in order to arrive at a suitable model for the mobile communications channel from which relevant statements for practical applications can be derived.<br>

Guenter at 16:37, 30 January 2023

2023-01-30T16:37:51Z

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	The general relationship between the four system functions is very complicated due to non-stationary effects.		The general relationship between the four system functions is very complicated due to non-stationary effects.

	[[File:~~EN_Mob_T_2_3_S2_neu~~.png\|right\|frame\|Connections between the description functions of the GWSSUS model '''KORREKTUR: PSD'''\|class=fit]]		[[File:EN_Mob_T_2_3_S2_v2.png\|right\|frame\|Connections between the description functions of the GWSSUS model '''KORREKTUR: PSD'''\|class=fit]]
	Compared to the general model, some limitations have to be made in order to arrive at a suitable model for the mobile communications channel from which relevant statements for practical applications can be derived.<br>		Compared to the general model, some limitations have to be made in order to arrive at a suitable model for the mobile communications channel from which relevant statements for practical applications can be derived.<br>

Guenter at 13:58, 29 January 2023

2023-01-29T13:58:36Z

Guenter at 13:54, 29 January 2023

2023-01-29T13:54:32Z

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	::<math>\varphi_{\rm VZ}(\Delta \tau, \Delta t) = \delta(\Delta \tau) \cdot {\it \Phi}_{\rm VZ}(\tau, \Delta t) \hspace{0.05cm}.</math>		::<math>\varphi_{\rm VZ}(\Delta \tau, \Delta t) = \delta(\Delta \tau) \cdot {\it \Phi}_{\rm VZ}(\tau, \Delta t) \hspace{0.05cm}.</math>

	== Power density ~~spectrum~~ of the time-variant impulse response==		== Power-spectral density of the time-variant impulse response==
	<br>~~'''KORREKTUR: power-spectral density'''~~		<br>
	[[File:P ID2170 Mob T 2 3 S3a v2.png\|right\|frame\|Delay's power-spectral density \|class=fit]]		[[File:P ID2170 Mob T 2 3 S3a v2.png\|right\|frame\|Delay's power-spectral density \|class=fit]]
	One obtains the   »'''delay's power-spectral density'''«   ${\it \Phi}_{\rm V}(\Delta \tau)$  by setting the second parameter of  ${\it \Phi}_{\rm VZ}(\Delta \tau, \Delta t)$  to  $\Delta t = 0$ .  The graphic on the right shows an exemplary curve.<br>		One obtains the   »'''delay's power-spectral density'''«   ${\it \Phi}_{\rm V}(\Delta \tau)$  by setting the second parameter of  ${\it \Phi}_{\rm VZ}(\Delta \tau, \Delta t)$  to  $\Delta t = 0$ .  The graphic on the right shows an exemplary curve.<br>
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	::<math>{\rm PDF}_{\rm V}(\tau) = \frac{{\it \Phi}_{\rm V}(\tau)}{\int_{0 }^{\infty}{\it \Phi}_{\rm V}(\tau)\hspace{0.15cm}{\rm d}\tau} \hspace{0.05cm}.</math>		::<math>{\rm PDF}_{\rm V}(\tau) = \frac{{\it \Phi}_{\rm V}(\tau)}{\int_{0 }^{\infty}{\it \Phi}_{\rm V}(\tau)\hspace{0.15cm}{\rm d}\tau} \hspace{0.05cm}.</math>

	''Note on nomenclature'':		'''Note on nomenclature''':
	*In the book  "Stochastic Signal Theory"  we would have denoted this  [[Theory_of_Stochastic_Signals/Probability_Density_Function_(PDF)#Definition_of_the_probability_density_function\|$\text{probability density function}$]]  with $f_\tau(\tau)$.		*In the book  "Stochastic Signal Theory"  we would have denoted this  [[Theory_of_Stochastic_Signals/Probability_Density_Function_(PDF)#Definition_of_the_probability_density_function\|$\text{probability density function}$]]  with $f_\tau(\tau)$.
	*To make the relation between  ${\it \Phi}_{\rm V}(\Delta \tau)$  and the PDF clear and to avoid confusion with the frequency $f$  we use the nomenclature given here.<br>		*To make the relation between  ${\it \Phi}_{\rm V}(\Delta \tau)$  and the PDF clear and to avoid confusion with the frequency $f$  we use the nomenclature given here.<br>
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	In the 1990s, the European Union founded the working group COST 207 with the aim to provide standardized channel models for cellular mobile communications.  where "COST" stands for  "European Cooperation in Science and Technology".<br>		In the 1990s, the European Union founded the working group COST 207 with the aim to provide standardized channel models for cellular mobile communications.  where "COST" stands for  "European Cooperation in Science and Technology".<br>

	In this international committee profiles for the delay time  $\tau$  have been developed, based on measurements and valid for different application scenarios.   In the following, four different delay's ~~power density spectra '''KORREKTUR:~~ power-spectral densities~~'''~~ are given, where the normalization factor  ${\it \Phi}_0 = {\it \Phi}_{\rm V}(\tau = 0)$  is always used.  The graph shows the ~~PDSs '''KORREKTUR:~~ PSDs~~'''~~of these profiles in logarithmic representation:		In this international committee profiles for the delay time  $\tau$  have been developed, based on measurements and valid for different application scenarios.   In the following, four different delay's power-spectral densities are given, where the normalization factor  ${\it \Phi}_0 = {\it \Phi}_{\rm V}(\tau = 0)$  is always used.  The graph shows the PSDs of these profiles in logarithmic representation:

	[[File:P ID2175 Mob T 2 3 S4a v1.png\|right\|frame\|Delay's ~~power density spectra '''KORREKTUR:~~ power-spectral densities~~'''~~ according to COST\|class=fit]]		[[File:P ID2175 Mob T 2 3 S4a v1.png\|right\|frame\|Delay's power-spectral densities according to COST\|class=fit]]

	'''(1)'''  profile  $\rm RA$  ("Rural Area"):		'''(1)'''  profile  $\rm RA$  ("Rural Area"):

Guenter at 13:49, 29 January 2023

2023-01-29T13:49:44Z

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	}}		}}

	== Generalized system functions of time variant systems ==		== Generalized system functions of time variant-systems ==
	<br>		<br>
	~~'''KORREKTUR: time-variant'''~~
	Linear time-invariant systems  $\rm (LTI)$  can be completely described with only two system functions,		Linear time-invariant systems  $\rm (LTI)$  can be completely described with only two system functions,
	*the transfer function  $H(f)$  and		*the transfer function  $H(f)$  and
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	*Additionally, the designations used in other chapters or literature are given  (gray letters).  In the other chapters these are also partly used.		*Additionally, the designations used in other chapters or literature are given  (gray letters).  In the other chapters these are also partly used.
	<br clear=all>		<br clear=all>
	${\rm (1)}$   At the top you can see the  »'''time variant impulse response'''« ~~'''KORREKTUR: time-variant'''~~   ${\eta}_{\rm VZ}(\tau,\hspace{0.05cm} t) \equiv h(\tau,\hspace{0.05cm} t)$  in the  "delay–time range".  The associated auto-correlation function  $\rm (ACF)$  is		${\rm (1)}$   At the top you can see the  »'''time-variant impulse response'''«   ${\eta}_{\rm VZ}(\tau,\hspace{0.05cm} t) \equiv h(\tau,\hspace{0.05cm} t)$  in the  "delay–time range".  The associated auto-correlation function  $\rm (ACF)$  is
	:::<math>\varphi_{\rm VZ}(\tau_1, t_1, \tau_2, t_2) = {\rm E} \big[ \eta_{\rm VZ}(\tau_1,\hspace{0.05cm} t_1) \cdot		:::<math>\varphi_{\rm VZ}(\tau_1, t_1, \tau_2, t_2) = {\rm E} \big[ \eta_{\rm VZ}(\tau_1,\hspace{0.05cm} t_1) \cdot
	\eta_{\rm VZ}^{\star}(\tau_2, t_2) \big]\hspace{0.05cm}. </math>		\eta_{\rm VZ}^{\star}(\tau_2, t_2) \big]\hspace{0.05cm}. </math>
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	\hspace{0.05cm}.</math>		\hspace{0.05cm}.</math>

	:The ACF of this ~~time variant '''KORREKTUR:~~ time-variant~~'''~~ transfer function is generally:		:The ACF of this time-variant transfer function is generally:

	:::<math>\varphi_{\rm FZ}(f_1, t_1, f_2, t_2) = {\rm E} \big [ \eta_{\rm FZ}(f_1, t_1) \cdot		:::<math>\varphi_{\rm FZ}(f_1, t_1, f_2, t_2) = {\rm E} \big [ \eta_{\rm FZ}(f_1, t_1) \cdot
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	{{BlaueBox\|TEXT=		{{BlaueBox\|TEXT=
	$\text{~~Hints~~:}$  ~~'''KORREKTUR: Notes'''~~		$\text{Notes:}$
	*The specified Fourier correlations between the system functions in the graph are illustrated by the outer, dark green arrows and are marked with   ${\rm F}_p\hspace{0.05cm}[\hspace{0.05cm} \cdot \hspace{0.05cm}]$ ,  <br>the index  $p$  indicates to which parameter  $\tau$,  $f$,  $t$  or  $f_{\rm D}$  does the Fourier transformation refer.		*The specified Fourier correlations between the system functions in the graph are illustrated by the outer, dark green arrows and are marked with   ${\rm F}_p\hspace{0.05cm}[\hspace{0.05cm} \cdot \hspace{0.05cm}]$ ,  <br>the index  $p$  indicates to which parameter  $\tau$,  $f$,  $t$  or  $f_{\rm D}$  does the Fourier transformation refer.

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	*The random process of the channel impulse response  $h(\tau,\hspace{0.1cm} t) = {\eta}_{\rm VZ}(\tau,\hspace{0.1cm} t)$  is generally assumed to be complex  (i.e., description in the equivalent low-pass range),  Gaussian  $($identifier  $\rm G)$  and zero-mean  (Rayleigh, not Rice, that means, no line of sight) .<br>		*The random process of the channel impulse response  $h(\tau,\hspace{0.1cm} t) = {\eta}_{\rm VZ}(\tau,\hspace{0.1cm} t)$  is generally assumed to be complex  (i.e., description in the equivalent low-pass range),  Gaussian  $($identifier  $\rm G)$  and zero-mean  (Rayleigh, not Rice, that means, no line of sight) .<br>

	*The random process is weakly stationary  ⇒   its characteristics change only slightly with time, and the ACF  $ {\varphi}_{\rm VZ}(\tau_1,\hspace{0.05cm} t_1,\hspace{0.05cm}\tau_2,\hspace{0.05cm} t_2)$  of the ~~time variant '''KORREKTUR:~~ time-variant~~'''~~ impulse response does not depend on the absolute times  $t_1$  and  $t_2$  but only on the time difference  $\Delta t = t_2 - t_1$.   This is indicated by the identifier  $\rm WSS$    ⇒   $\rm W$ide $\rm S$ense $\rm S$tationary.<br>		*The random process is weakly stationary  ⇒   its characteristics change only slightly with time, and the ACF  $ {\varphi}_{\rm VZ}(\tau_1,\hspace{0.05cm} t_1,\hspace{0.05cm}\tau_2,\hspace{0.05cm} t_2)$  of the time-variant impulse response does not depend on the absolute times  $t_1$  and  $t_2$  but only on the time difference  $\Delta t = t_2 - t_1$.   This is indicated by the identifier  $\rm WSS$    ⇒   $\rm W$ide $\rm S$ense $\rm S$tationary.<br>

	*The individual echoes due to multipath propagation are uncorrelated, which is expressed by the identifier  $\rm US$   ⇒   $\rm U$ncorrelated $\rm S$cattering</i>.		*The individual echoes due to multipath propagation are uncorrelated, which is expressed by the identifier  $\rm US$   ⇒   $\rm U$ncorrelated $\rm S$cattering</i>.
	<br clear=all>		<br clear=all>
	The mobile communications channel can be described in full according to this graph.  The individual ~~power density spectra '''KORREKTUR:~~ power-spectral densities~~'''~~  (labeled blue)  and the correlation function  (labeled red)  is explained in detail in the following sections.<br>		The mobile communications channel can be described in full according to this graph.  The individual power-spectral densities  (labeled blue)  and the correlation function  (labeled red)  is explained in detail in the following sections.<br>


	== ~~Autocorrelation~~ function of the time-variant impulse response==		== Auto-correlation function of the time-variant impulse response==
	<br>		<br>
	We now consider the  [[Theory_of_Stochastic_Signals/Auto-Correlation_Function_(ACF)\|$\text{auto-correlation function}$]]  $\rm (ACF)$  of the ~~time variant '''KORREKTUR:~~ time-variant~~'''~~ impulse response   ⇒   $h(\tau,\hspace{0.1cm} t) = {\eta}_{\rm VZ}(\tau,\hspace{0.1cm} t)$  more closely.  It shows:		We now consider the  [[Theory_of_Stochastic_Signals/Auto-Correlation_Function_(ACF)\|$\text{auto-correlation function}$]]  $\rm (ACF)$  of the time-variant impulse response   ⇒   $h(\tau,\hspace{0.1cm} t) = {\eta}_{\rm VZ}(\tau,\hspace{0.1cm} t)$  more closely.  It shows:

	*Based on the  $\rm WSS$ property, the auto-correlation function can be written with  $\Delta t = t_2 - t_1$ :		*Based on the  $\rm WSS$ property, the auto-correlation function can be written with  $\Delta t = t_2 - t_1$ :

Hwang at 15:19, 26 January 2023

2023-01-26T15:19:16Z

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	<br clear=all>		<br clear=all>

	==Exercises to the chapter==		==Exercises for the chapter==
	<br>		<br>
	[[Aufgaben:Exercise 2.5: Scatter Function]]		[[Aufgaben:Exercise 2.5: Scatter Function]]

Hwang at 11:25, 26 January 2023

2023-01-26T11:25:04Z

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	{{GraueBox\|TEXT=		{{GraueBox\|TEXT=
	$\text{Example 3:}$  The diagram is valid for a time–variant channel without direct component.  Shown on the left is the  [[Mobile_Communications/~~Statistical_bonds_within_the Rayleigh process~~#~~ACF_and_PDS_with_Rayleigh_fading~~\|$\text{Jakes spectrum}$]]  ${\it \Phi}_{\rm D}(f_{\rm D})$.		$\text{Example 3:}$  The diagram is valid for a time–variant channel without direct component.  Shown on the left is the  [[Mobile_Communications/Statistical_Bindings_within_the_Rayleigh_Process#ACF_and_PSD_with_Rayleigh.E2.80.93Fading\|$\text{Jakes spectrum}$]]  ${\it \Phi}_{\rm D}(f_{\rm D})$.
	[[File:P ID2181 Mob T 2 3 S8b v1.png\|right\|frame\|Doppler spread and correlation time\|class=fit]]		[[File:P ID2181 Mob T 2 3 S8b v1.png\|right\|frame\|Doppler spread and correlation time\|class=fit]]
	The Doppler spread  $B_{\rm D}$  can be determined from this:		The Doppler spread  $B_{\rm D}$  can be determined from this:

Hwang at 11:13, 26 January 2023

2023-01-26T11:13:41Z

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	*the  [[Mobile_Communications/The_GWSSUS_Channel_Model#Autocorrelation_function_of_the_time-variant_impulse_response\|$\text{delay–time's cross-power-spectral density}$]]:		*the  [[Mobile_Communications/The_GWSSUS_Channel_Model#Autocorrelation_function_of_the_time-variant_impulse_response\|$\text{delay–time's cross-power-spectral density}$]]:
	:$${\it \Phi}_{\rm VZ}(\tau, \Delta t)\hspace{0.55cm}\Rightarrow \hspace{0.3cm}\text{with} \hspace{0.2cm}\Delta t = 0\text{:} \hspace{0.2cm} {\it \Phi}_{\rm V}(\tau),$$		:$${\it \Phi}_{\rm VZ}(\tau, \Delta t)\hspace{0.55cm}\Rightarrow \hspace{0.3cm}\text{with} \hspace{0.2cm}\Delta t = 0\text{:} \hspace{0.2cm} {\it \Phi}_{\rm V}(\tau),$$
	*the  [[Mobile_Communications/The_GWSSUS_Channel_Model#~~ACF_and_PDS_of_the_frequency~~-variant_transfer_function\|$\text{frequency–Doppler's cross-power-spectral density}$]]:		*the  [[Mobile_Communications/The_GWSSUS_Channel_Model#ACF_and_PSD_of_the_frequency-variant_transfer_function\|$\text{frequency–Doppler's cross-power-spectral density}$]]:
	:$${\it \Phi}_{\rm FD}(\Delta f, f_{\rm D})\hspace{0.3cm}\Rightarrow \hspace{0.3cm}\text{with} \hspace{0.2cm}\Delta f = 0\text{:} \hspace{0.2cm} {\it \Phi}_{\rm D}( f_{\rm D}),$$		:$${\it \Phi}_{\rm FD}(\Delta f, f_{\rm D})\hspace{0.3cm}\Rightarrow \hspace{0.3cm}\text{with} \hspace{0.2cm}\Delta f = 0\text{:} \hspace{0.2cm} {\it \Phi}_{\rm D}( f_{\rm D}),$$
	*the  [[Mobile_Communications/The_GWSSUS_Channel_Model#~~ACF_and_PDS_of_the_delay~~-Doppler_function \|$\text{delay–Doppler's cross-power-spectral density}$ ]]:		*the  [[Mobile_Communications/The_GWSSUS_Channel_Model#ACF_and_PSD_of_the_delay-Doppler_function \|$\text{delay–Doppler's cross-power-spectral density}$ ]]:
	:$${\it \Phi}_{\rm VD}(\tau, f_{\rm D})= {\it \Phi}_{\rm V}(\tau) \cdot {\it \Phi}_{\rm D}(f_{\rm D})\hspace{0.05cm}.$$		:$${\it \Phi}_{\rm VD}(\tau, f_{\rm D})= {\it \Phi}_{\rm V}(\tau) \cdot {\it \Phi}_{\rm D}(f_{\rm D})\hspace{0.05cm}.$$
	<br clear=all>		<br clear=all>

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	<br>         According to the graph, this can be reached in two ways:		<br>         According to the graph, this can be reached in two ways:

	::$$\eta_{\rm FD}(f, f_{\rm D})		::$$\eta_{\rm FD}(f, f_{\rm D})\hspace{0.2cm} \stackrel{f_{\rm D}, \hspace{0.05cm}t}{\bullet\!\!-\!\!\!-\!\!\!-\!\!\circ} \hspace{0.2cm} \eta_{\rm FZ}(f, t)\hspace{0.05cm},\hspace{0.5cm}\eta_{\rm FD}(f, f_{\rm D})\hspace{0.2cm} \stackrel{f, \hspace{0.05cm}\tau}{\bullet\!\!-\!\!\!-\!\!\!-\!\!\circ} \hspace{0.2cm} \eta_{\rm VD}(\tau, f_{\rm D})\hspace{0.05cm}.$$
	\hspace{0.2cm} \stackrel{f_{\rm D}, \hspace{0.05cm}t}{\bullet\!\!-\!\!\!-\!\!\!-\!\!\circ} \hspace{0.2cm} \eta_{\rm FZ}(f, t)\hspace{0.05cm},\hspace{0.5cm}\eta_{\rm FD}(f, f_{\rm D})
	\hspace{0.2cm} \stackrel{f, \hspace{0.05cm}\tau}{\bullet\!\!-\!\!\!-\!\!\!-\!\!\circ} \hspace{0.2cm} \eta_{\rm VD}(\tau, f_{\rm D})\hspace{0.05cm}.$$

	{{BlaueBox\|TEXT=		{{BlaueBox\|TEXT=
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	*The specified Fourier correlations between the system functions in the graph are illustrated by the outer, dark green arrows and are marked with   ${\rm F}_p\hspace{0.05cm}[\hspace{0.05cm} \cdot \hspace{0.05cm}]$ ,  <br>the index  $p$  indicates to which parameter  $\tau$,  $f$,  $t$  or  $f_{\rm D}$  does the Fourier transformation refer.		*The specified Fourier correlations between the system functions in the graph are illustrated by the outer, dark green arrows and are marked with   ${\rm F}_p\hspace{0.05cm}[\hspace{0.05cm} \cdot \hspace{0.05cm}]$ ,  <br>the index  $p$  indicates to which parameter  $\tau$,  $f$,  $t$  or  $f_{\rm D}$  does the Fourier transformation refer.

	*The inner  (lighter)  arrows indicate the links via the  [[Signal_Representation/~~Fourier_Transform_and_its_Inverse~~#The_second_Fourier_integral\|$\text{inverse Fourier transform}$]].   For this we use the notation  ${ {\rm F}_p}^{-1}\hspace{0.05cm}[ \hspace{0.05cm} \cdot \hspace{0.05cm} ]$.		*The inner  (lighter)  arrows indicate the links via the  [[Signal_Representation/The_Fourier_Transform_and_its_Inverse#The_second_Fourier_integral\|$\text{inverse Fourier transform}$]].   For this we use the notation  ${ {\rm F}_p}^{-1}\hspace{0.05cm}[ \hspace{0.05cm} \cdot \hspace{0.05cm} ]$.

	*The applet  [[Applets:Impulses and Spectra\|"Impulses and Spectra"]] illustrates the connection between the time and frequency domain, which can be described by formulas using Fourier transformation and Fourier inverse transformation.}}		*The applet  [[Applets:Impulses and Spectra\|"Impulses and Spectra"]] illustrates the connection between the time and frequency domain, which can be described by formulas using Fourier transformation and Fourier inverse transformation.}}
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	*With this approach, the auto-correlation function  $\varphi_{\rm VZ}(\Delta \tau, \Delta t)$  and the power-spectral density   ${\it \Phi}_{\rm VZ}(\tau, \Delta t) $  are not connected via the Fourier transform as usual, but are linked via a Dirac delta function:		*With this approach, the auto-correlation function  $\varphi_{\rm VZ}(\Delta \tau, \Delta t)$  and the power-spectral density   ${\it \Phi}_{\rm VZ}(\tau, \Delta t) $  are not connected via the Fourier transform as usual, but are linked via a Dirac delta function:
	::<math>\varphi_{\rm VZ}(\Delta \tau, \Delta t) = \delta(\Delta \tau) \cdot {\it \Phi}_{\rm VZ}(\tau, \Delta t) \hspace{0.05cm}. </math>		::<math>\varphi_{\rm VZ}(\Delta \tau, \Delta t) = \delta(\Delta \tau) \cdot {\it \Phi}_{\rm VZ}(\tau, \Delta t) \hspace{0.05cm}. </math>
	*Not all symmetry properties that follow from the  [[Theory_of_Stochastic_Signals/~~Power_Density_Spectrum_(PDS)~~#Wiener-Khintchine_Theorem\| $\text{Wiener–Khintchine theorem}$]]  are thus given here.  In particular it is quite possible and even very likely that such a power-spectral density is an odd function.}}<br>		*Not all symmetry properties that follow from the  [[Theory_of_Stochastic_Signals/Power-Spectral_Density#Wiener-Khintchine_Theorem\| $\text{Wiener–Khintchine theorem}$]]  are thus given here.  In particular it is quite possible and even very likely that such a power-spectral density is an odd function.}}<br>

	In the overview in the last section, the  »'''delay–time's cross power-spectral density'''«  ${\it \Phi}_{\rm VZ}(\tau, \Delta t) $  can be seen in the top middle.		In the overview in the last section, the  »'''delay–time's cross power-spectral density'''«  ${\it \Phi}_{\rm VZ}(\tau, \Delta t) $  can be seen in the top middle.
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	== ACF and PSD of the frequency-variant transfer function==		== ACF and PSD of the frequency-variant transfer function==
	<br>		<br>
	The system function   $\eta_{\rm FD}(f, f_{\rm D})$  described in the  [[Mobile_Communications/~~The_GWSSUS channel model~~#~~Generalized system functions_of time variant_systems~~\|"overview in the first section of this chapter"]]  is also known as the  "Frequency-variant Transfer Function"  where the adjective  "frequency-variant"  refers to the Doppler frequency $f_{\rm D}$.		The system function   $\eta_{\rm FD}(f, f_{\rm D})$  described in the  [[Mobile_Communications/The_GWSSUS_Channel_Model#Generalized_system_functions_of_time_variant-systems\|"overview in the first section of this chapter"]]  is also known as the  "Frequency-variant Transfer Function"  where the adjective  "frequency-variant"  refers to the Doppler frequency $f_{\rm D}$.

	The associated auto-correlation function  $\rm (ACF)$  is defined as follows:		The associated auto-correlation function  $\rm (ACF)$  is defined as follows:
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	== ACF and PSD of the delay-Doppler function ==		== ACF and PSD of the delay-Doppler function ==
	<br>		<br>
	The system function shown in the  [[Mobile_Communications/~~The_GWSSUS channel model~~#~~Generalized_system functions_of time variant_systems~~\|"overview in the first section of this chapter"]]  on the left hand side was named  $\eta_{\rm VD}(\tau, f_{\rm D})$ .		The system function shown in the  [[Mobile_Communications/The_GWSSUS_Channel_Model#Generalized_system_functions_of_time_variant-systems\|"overview in the first section of this chapter"]]  on the left hand side was named  $\eta_{\rm VD}(\tau, f_{\rm D})$ .

	The auto-correlation function  $\rm (ACF)$  of this delay–Doppler function can be written with  $\Delta \tau = \tau_2 - \tau_1$  and  $\Delta f_{\rm D} = f_{\rm D2} - f_{\rm D1}$  taking into account the GWSSUS properties with  $\Delta \tau = \tau_2 - \tau_1$  and  $\Delta f_{\rm D} = f_{\rm D2} - f_{\rm D1}$  as follows		The auto-correlation function  $\rm (ACF)$  of this delay–Doppler function can be written with  $\Delta \tau = \tau_2 - \tau_1$  and  $\Delta f_{\rm D} = f_{\rm D2} - f_{\rm D1}$  taking into account the GWSSUS properties with  $\Delta \tau = \tau_2 - \tau_1$  and  $\Delta f_{\rm D} = f_{\rm D2} - f_{\rm D1}$  as follows
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	*If you set the parameters  $\Delta t = 0$  or  $\Delta f = 0$  in this two–dimensional function, the separate correlation functions for the frequency domain or the time domain result:		*If you set the parameters  $\Delta t = 0$  or  $\Delta f = 0$  in this two–dimensional function, the separate correlation functions for the frequency domain or the time domain result:

	:$$\varphi_{\rm F}(\Delta f) = \varphi_{\rm FZ}(\Delta f, \Delta t = 0) \hspace{0.05cm},\hspace{0.5cm}		:$$\varphi_{\rm F}(\Delta f) = \varphi_{\rm FZ}(\Delta f, \Delta t = 0) \hspace{0.05cm},\hspace{0.5cm}\varphi_{\rm Z}(\Delta t) = \varphi_{\rm FZ}(\Delta f = 0, \Delta t ) \hspace{0.05cm}.$$
	\varphi_{\rm Z}(\Delta t) = \varphi_{\rm FZ}(\Delta f = 0, \Delta t ) \hspace{0.05cm}.$$

	*From the graph it is also clear that these correlation functions correspond to the derived power-spectral densities via the Fourier transform:		*From the graph it is also clear that these correlation functions correspond to the derived power-spectral densities via the Fourier transform:

← Older revision		Revision as of 15:58, 29 January 2023
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	*The first argument  $\Delta f = f_2 - f_1$  takes into account that ACF and PSD depend only on the frequency difference due to the  "stationarity" .		*The first argument  $\Delta f = f_2 - f_1$  takes into account that ACF and PSD depend only on the frequency difference due to the  "stationarity" .

	*The factor  $\delta (\Delta f_{\rm D})$  with  $\Delta f_{\rm D} = f_{\rm D_2} - f_{\rm D_1}$  expresses the  "uncorrelation of the ~~PDS '''KORREKTUR:~~ PSD~~'''~~"  with respect to the Doppler shift.<br>		*The factor  $\delta (\Delta f_{\rm D})$  with  $\Delta f_{\rm D} = f_{\rm D_2} - f_{\rm D_1}$  expresses the  "uncorrelation of the PSD"  with respect to the Doppler shift.<br>

	*You get from  ${\it \Phi}_{\rm FD}(\Delta f, f_{\rm D})$  to  »'''Doppler's power-spectral density'''«   ${\it \Phi}_{\rm D}(f_{\rm D})$  if you set  $\Delta f= 0$.		*You get from  ${\it \Phi}_{\rm FD}(\Delta f, f_{\rm D})$  to  »'''Doppler's power-spectral density'''«   ${\it \Phi}_{\rm D}(f_{\rm D})$  if you set  $\Delta f= 0$.
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	*Both, the system function  $\eta_{\rm VD}(\tau, f_{\rm D})$  and the derived functions  $\varphi _{\rm VD}(\Delta \tau, \Delta f_{\rm D})$  and  ${\it \Phi}_{\rm VD}(\tau, f_{\rm D})$,  are dimensionless.   For more information on this,  see the specification for  [[Aufgaben:Exercise 2.6: Dimensions in GWSSUS\|"Exercise 2.6"]].		*Both, the system function  $\eta_{\rm VD}(\tau, f_{\rm D})$  and the derived functions  $\varphi _{\rm VD}(\Delta \tau, \Delta f_{\rm D})$  and  ${\it \Phi}_{\rm VD}(\tau, f_{\rm D})$,  are dimensionless.   For more information on this,  see the specification for  [[Aufgaben:Exercise 2.6: Dimensions in GWSSUS\|"Exercise 2.6"]].

	*Furthermore, if the GWSSUS requirements are met, the scatter function is equal to the product of the delay's and Doppler's ~~PDSs '''KORREKTUR:~~ PSDs~~'''~~:		*Furthermore, if the GWSSUS requirements are met, the scatter function is equal to the product of the delay's and Doppler's PSDs:

	::<math>{\it \Phi}_{\rm VD}(\tau, f_{\rm D}) = {\it \Phi}_{\rm V}(\tau) \cdot {\it \Phi}_{\rm D}(f_{\rm D})\hspace{0.05cm}.</math>		::<math>{\it \Phi}_{\rm VD}(\tau, f_{\rm D}) = {\it \Phi}_{\rm V}(\tau) \cdot {\it \Phi}_{\rm D}(f_{\rm D})\hspace{0.05cm}.</math>
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	== ACF and PSD of the time-variant transfer function ==		== ACF and PSD of the time-variant transfer function ==
	<br>		<br>
	The following diagram shows all the relationships between the individual ~~power density spectra '''KORREKTUR:~~ power-spectral densities~~'''~~ once again in compact form.		The following diagram shows all the relationships between the individual power-spectral densities once again in compact form.
	[[File:P ID2176 Mob T 2 3 S7 v1.png\|right\|frame\|Compact summary of all GWSSUS description quantities \|class=fit]]		[[File:P ID2176 Mob T 2 3 S7 v1.png\|right\|frame\|Compact summary of all GWSSUS description quantities \|class=fit]]
	This has already been discussed in the last sections:		This has already been discussed in the last sections:
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	\varphi_{\rm Z}(\Delta t) = \varphi_{\rm FZ}(\Delta f = 0, \Delta t ) \hspace{0.05cm}.$$		\varphi_{\rm Z}(\Delta t) = \varphi_{\rm FZ}(\Delta f = 0, \Delta t ) \hspace{0.05cm}.$$

	*From the graph it is also clear that these correlation functions correspond to the derived ~~power spectral '''KORREKTUR:~~ power-spectral~~'''~~ densities via the Fourier transform:		*From the graph it is also clear that these correlation functions correspond to the derived power-spectral densities via the Fourier transform:

	::<math>\varphi_{\rm F}(\Delta f) \hspace{0.2cm} {\bullet\!\!-\!\!\!-\!\!\!-\!\!\circ} \hspace{0.2cm} {\it \Phi}_{\rm V}(\tau)\hspace{0.05cm},		::<math>\varphi_{\rm F}(\Delta f) \hspace{0.2cm} {\bullet\!\!-\!\!\!-\!\!\!-\!\!\circ} \hspace{0.2cm} {\it \Phi}_{\rm V}(\tau)\hspace{0.05cm},
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	By suitable normalization to the respective area  $1$  the density functions result with respect to the delay time  $\tau$  or the Doppler frequency $f_{\rm D}$.<br>		By suitable normalization to the respective area  $1$  the density functions result with respect to the delay time  $\tau$  or the Doppler frequency $f_{\rm D}$.<br>

	Characteristic values can be derived from the ~~power spectral '''KORREKTUR:~~ power-spectral~~'''~~ densities or the corresponding correlation functions.  The most important ones are listed here:		Characteristic values can be derived from the power-spectral densities or the corresponding correlation functions.  The most important ones are listed here:

	{{BlaueBox\|TEXT=		{{BlaueBox\|TEXT=
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	[[File:P ID2183 Mob T 2 3 S9a.png\|right\|frame\|~~Time variant '''KORREKTUR:~~ Time-variant~~'''~~ transfer function  $($the absolute value squared is  simulated$)$]]		[[File:P ID2183 Mob T 2 3 S9a.png\|right\|frame\|Time-variant transfer function  $($the absolute value squared is  simulated$)$]]
	{{GraueBox\|TEXT=		{{GraueBox\|TEXT=
	$\text{Example 4:}$  The graphic from  [Hin08]<ref name='Hin08'>Hindelang, T.:  Mobile Communications.		$\text{Example 4:}$  The graphic from  [Hin08]<ref name='Hin08'>Hindelang, T.:  Mobile Communications.