Mobile Communications/The GWSSUS Channel Model - Revision history

Guenter at 16:38, 30 January 2023

2023-01-30T16:38:16Z

Guenter at 16:37, 30 January 2023

2023-01-30T16:37:51Z

Guenter at 13:58, 29 January 2023

2023-01-29T13:58:36Z

Guenter at 13:54, 29 January 2023

2023-01-29T13:54:32Z

Guenter at 13:49, 29 January 2023

2023-01-29T13:49:44Z

Hwang at 15:19, 26 January 2023

2023-01-26T15:19:16Z

Hwang at 11:25, 26 January 2023

2023-01-26T11:25:04Z

Hwang at 11:13, 26 January 2023

2023-01-26T11:13:41Z

Hwang at 11:12, 26 January 2023

2023-01-26T11:12:08Z

Hwang at 11:01, 26 January 2023

2023-01-26T11:01:58Z

@@ Line 76: / Line 76: @@
 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>

@@ Line 76: / Line 76: @@
 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>

@@ Line 206: / Line 206: @@
 *The first argument&nbsp; $\Delta f = f_2 - f_1$&nbsp; takes into account that ACF and PSD depend only on the frequency difference due to the&nbsp; "stationarity"&nbsp;.
-*The factor&nbsp; $\delta (\Delta f_{\rm D})$&nbsp; with&nbsp; $\Delta f_{\rm D} = f_{\rm D_2} - f_{\rm D_1}$&nbsp; expresses the&nbsp; "uncorrelation of the PDS '''KORREKTUR: PSD'''"&nbsp; with respect to the Doppler shift.<br>
+*The factor&nbsp; $\delta (\Delta f_{\rm D})$&nbsp; with&nbsp; $\Delta f_{\rm D} = f_{\rm D_2} - f_{\rm D_1}$&nbsp; expresses the&nbsp; "uncorrelation of the PSD"&nbsp; with respect to the Doppler shift.<br>
 *You get from&nbsp; ${\it \Phi}_{\rm FD}(\Delta f, f_{\rm D})$&nbsp; to&nbsp; &raquo;'''Doppler's power-spectral density'''&laquo; &nbsp; ${\it \Phi}_{\rm D}(f_{\rm D})$&nbsp; if you set&nbsp; $\Delta f= 0$.&nbsp;
@@ Line 246: / Line 246: @@
 *Both, the system function&nbsp; $\eta_{\rm VD}(\tau, f_{\rm D})$&nbsp; and the derived functions&nbsp; $\varphi _{\rm VD}(\Delta \tau, \Delta f_{\rm D})$&nbsp; and&nbsp; ${\it \Phi}_{\rm VD}(\tau, f_{\rm D})$,&nbsp; are dimensionless. &nbsp; For more information on this,&nbsp; see the specification for&nbsp; [[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>
@@ Line 265: / Line 265: @@
 == ACF and PSD of the time-variant transfer function ==
 <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]]
 This has already been discussed in the last sections:
@@ Line 301: / Line 301: @@
 \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},
@@ Line 315: / Line 315: @@
 By suitable normalization to the respective area&nbsp; $1$&nbsp; the density functions result with respect to the delay time&nbsp; $\tau$&nbsp; 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.&nbsp; The most important ones are listed here:
+Characteristic values can be derived from the power-spectral  densities or the corresponding correlation functions.&nbsp; The most important ones are listed here:
 {{BlaueBox|TEXT=
@@ Line 408: / Line 408: @@
-[[File:P ID2183 Mob T 2 3 S9a.png|right|frame|Time variant '''KORREKTUR: Time-variant''' transfer function&nbsp; $($the absolute value squared is&nbsp; simulated$)$]]
+[[File:P ID2183 Mob T 2 3 S9a.png|right|frame|Time-variant  transfer function&nbsp; $($the absolute value squared is&nbsp; simulated$)$]]
 {{GraueBox|TEXT=
 $\text{Example 4:}$&nbsp; The graphic from&nbsp; [Hin08]<ref name='Hin08'>Hindelang, T.:&nbsp; Mobile Communications.&nbsp;

@@ Line 127: / Line 127: @@
 ::<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]]
 One obtains the &nbsp; &raquo;'''delay's power-spectral density'''&laquo; &nbsp; ${\it \Phi}_{\rm V}(\Delta \tau)$&nbsp; by setting the second parameter of&nbsp; ${\it \Phi}_{\rm VZ}(\Delta \tau, \Delta t)$&nbsp; to&nbsp; $\Delta t = 0$&nbsp;.&nbsp; The graphic on the right shows an exemplary curve.<br>
@@ Line 143: / Line 143: @@
 ::<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&nbsp; "Stochastic Signal Theory"&nbsp; we would have denoted this&nbsp; [[Theory_of_Stochastic_Signals/Probability_Density_Function_(PDF)#Definition_of_the_probability_density_function|$\text{probability density function}$]]&nbsp; with $f_\tau(\tau)$.
 *To make the relation between&nbsp; ${\it \Phi}_{\rm V}(\Delta \tau)$&nbsp; and the PDF clear and to avoid confusion with the frequency $f$&nbsp; we use the nomenclature given here.<br>
@@ Line 154: / Line 154: @@
 In the 1990s, the European Union founded the working group COST 207 with the aim to provide standardized channel models for cellular mobile communications.&nbsp; where "COST" stands for&nbsp; "European Cooperation in Science and Technology".<br>
-In this international committee profiles for the delay time&nbsp; $\tau$&nbsp; have been developed, based on measurements and valid for different application scenarios. &nbsp; In the following, four different delay's power density spectra '''KORREKTUR: power-spectral densities''' are given, where the normalization factor&nbsp; ${\it \Phi}_0 = {\it \Phi}_{\rm V}(\tau = 0)$&nbsp; is always used.&nbsp; The graph shows the PDSs '''KORREKTUR: PSDs'''of these profiles in logarithmic representation:
+In this international committee profiles for the delay time&nbsp; $\tau$&nbsp; have been developed, based on measurements and valid for different application scenarios. &nbsp; In the following, four different delay's power-spectral densities are given, where the normalization factor&nbsp; ${\it \Phi}_0 = {\it \Phi}_{\rm V}(\tau = 0)$&nbsp; is always used.&nbsp; 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)'''&nbsp; profile&nbsp; $\rm RA$&nbsp; ("Rural Area"):

@@ Line 6: / Line 6: @@
 }}
-== Generalized system functions of time variant systems ==
+== Generalized system functions of time variant-systems ==
 <br>
 Linear time-invariant systems&nbsp; $\rm (LTI)$&nbsp; can be completely described with only two system functions,
 *the transfer function&nbsp; $H(f)$&nbsp; and
@@ Line 32: / Line 31: @@
 *Additionally, the designations used in other chapters or literature are given&nbsp; (gray letters).&nbsp; In the other chapters these are also partly used.
 <br clear=all>
-${\rm (1)}$ &nbsp; At the top you can see the&nbsp; &raquo;'''time variant impulse response'''&laquo; '''KORREKTUR: time-variant''' &nbsp; ${\eta}_{\rm VZ}(\tau,\hspace{0.05cm} t) \equiv h(\tau,\hspace{0.05cm} t)$&nbsp; in the&nbsp; "delay&ndash;time range".&nbsp; The associated auto-correlation function&nbsp; $\rm (ACF)$&nbsp; is
+${\rm (1)}$ &nbsp; At the top you can see the&nbsp; &raquo;'''time-variant impulse response'''&laquo;  &nbsp; ${\eta}_{\rm VZ}(\tau,\hspace{0.05cm} t) \equiv h(\tau,\hspace{0.05cm} t)$&nbsp; in the&nbsp; "delay&ndash;time range".&nbsp; The associated auto-correlation function&nbsp; $\rm (ACF)$&nbsp; 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
   \eta_{\rm VZ}^{\star}(\tau_2, t_2) \big]\hspace{0.05cm}. </math>
@@ Line 43: / Line 42: @@
   \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
@@ Line 65: / Line 64: @@
 {{BlaueBox|TEXT=
-$\text{Hints:}$&nbsp; '''KORREKTUR: Notes'''
+$\text{Notes:}$&nbsp;
 *The specified Fourier correlations between the system functions in the graph are illustrated by the outer, dark green arrows and are marked with &nbsp; ${\rm F}_p\hspace{0.05cm}[\hspace{0.05cm} \cdot \hspace{0.05cm}]$&nbsp;,&nbsp; <br>the index &nbsp;$p$&nbsp; indicates to which parameter&nbsp; $\tau$,&nbsp; $f$,&nbsp; $t$&nbsp; or&nbsp; $f_{\rm D}$&nbsp; does the  Fourier transformation refer.
@@ Line 83: / Line 82: @@
 *The random process of the channel impulse response&nbsp; $h(\tau,\hspace{0.1cm} t) = {\eta}_{\rm VZ}(\tau,\hspace{0.1cm} t)$&nbsp; is generally assumed to be complex&nbsp; (i.e., description in the equivalent low-pass range),&nbsp; Gaussian&nbsp; $($identifier&nbsp; $\rm G)$&nbsp; and zero-mean&nbsp; (Rayleigh, not Rice, that means, no line of sight)&nbsp;.<br>
-*The random process is weakly stationary&nbsp; &rArr; &nbsp; its characteristics change only slightly with time, and the ACF&nbsp; $ {\varphi}_{\rm VZ}(\tau_1,\hspace{0.05cm} t_1,\hspace{0.05cm}\tau_2,\hspace{0.05cm} t_2)$&nbsp; of the time variant '''KORREKTUR: time-variant''' impulse response does not depend on the absolute times&nbsp; $t_1$&nbsp; and&nbsp; $t_2$&nbsp; but only on the time difference&nbsp; $\Delta t = t_2 - t_1$. &nbsp; This is indicated by the identifier&nbsp; $\rm WSS$&nbsp; &nbsp;&nbsp;&#8658;&nbsp;&nbsp; $\rm W$ide $\rm S$ense $\rm S$tationary.<br>
+*The random process is weakly stationary&nbsp; &rArr; &nbsp; its characteristics change only slightly with time, and the ACF&nbsp; $ {\varphi}_{\rm VZ}(\tau_1,\hspace{0.05cm} t_1,\hspace{0.05cm}\tau_2,\hspace{0.05cm} t_2)$&nbsp; of the time-variant impulse response does not depend on the absolute times&nbsp; $t_1$&nbsp; and&nbsp; $t_2$&nbsp; but only on the time difference&nbsp; $\Delta t = t_2 - t_1$. &nbsp; This is indicated by the identifier&nbsp; $\rm WSS$&nbsp; &nbsp;&nbsp;&#8658;&nbsp;&nbsp; $\rm W$ide $\rm S$ense $\rm S$tationary.<br>
 *The individual echoes due to multipath propagation are uncorrelated, which is expressed by the identifier&nbsp; $\rm US$ &nbsp;&nbsp;&#8658;&nbsp;&nbsp; $\rm U$ncorrelated $\rm S$cattering</i>.
 <br clear=all>
-The mobile communications channel can be described in full according to this graph.&nbsp; The individual power density spectra '''KORREKTUR: power-spectral densities'''&nbsp; (labeled blue)&nbsp; and the correlation function&nbsp; (labeled red)&nbsp; is explained in detail in the following sections.<br>
+The mobile communications channel can be described in full according to this graph.&nbsp; The individual  power-spectral densities&nbsp; (labeled blue)&nbsp; and the correlation function&nbsp; (labeled red)&nbsp; 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>
-We now consider the&nbsp; [[Theory_of_Stochastic_Signals/Auto-Correlation_Function_(ACF)|$\text{auto-correlation function}$]]&nbsp; $\rm (ACF)$&nbsp; of the time variant '''KORREKTUR: time-variant''' impulse response &nbsp; &#8658; &nbsp; $h(\tau,\hspace{0.1cm} t) = {\eta}_{\rm VZ}(\tau,\hspace{0.1cm} t)$&nbsp; more closely.&nbsp; It shows:
+We now consider the&nbsp; [[Theory_of_Stochastic_Signals/Auto-Correlation_Function_(ACF)|$\text{auto-correlation function}$]]&nbsp; $\rm (ACF)$&nbsp; of the time-variant impulse response &nbsp; &#8658; &nbsp; $h(\tau,\hspace{0.1cm} t) = {\eta}_{\rm VZ}(\tau,\hspace{0.1cm} t)$&nbsp; more closely.&nbsp; It shows:
 *Based on the&nbsp; $\rm WSS$ property, the auto-correlation function can be written with&nbsp; $\Delta t = t_2 - t_1$&nbsp;:

@@ Line 428: / Line 428: @@
 <br clear=all>
-==Exercises to the chapter==
+==Exercises for the chapter==
 <br>
 [[Aufgaben:Exercise 2.5: Scatter Function]]

@@ Line 375: / Line 375: @@
 {{GraueBox|TEXT=
-$\text{Example 3:}$&nbsp; The diagram is valid for a time&ndash;variant channel without direct component.&nbsp; Shown on the left is the&nbsp; [[Mobile_Communications/Statistical_bonds_within_the Rayleigh process#ACF_and_PDS_with_Rayleigh_fading|$\text{Jakes spectrum}$]]&nbsp; ${\it \Phi}_{\rm D}(f_{\rm D})$.
+$\text{Example 3:}$&nbsp; The diagram is valid for a time&ndash;variant channel without direct component.&nbsp; Shown on the left is the&nbsp; [[Mobile_Communications/Statistical_Bindings_within_the_Rayleigh_Process#ACF_and_PSD_with_Rayleigh.E2.80.93Fading|$\text{Jakes spectrum}$]]&nbsp; ${\it \Phi}_{\rm D}(f_{\rm D})$.
 [[File:P ID2181 Mob T 2 3 S8b v1.png|right|frame|Doppler spread and correlation time|class=fit]]
 The Doppler spread&nbsp; $B_{\rm D}$&nbsp; can be determined from this:

@@ Line 271: / Line 271: @@
 *the&nbsp; [[Mobile_Communications/The_GWSSUS_Channel_Model#Autocorrelation_function_of_the_time-variant_impulse_response|$\text{delay&ndash;time's cross-power-spectral density}$]]:
 :$${\it \Phi}_{\rm VZ}(\tau, \Delta t)\hspace{0.55cm}\Rightarrow \hspace{0.3cm}\text{with}  \hspace{0.2cm}\Delta t = 0\text{:}  \hspace{0.2cm} {\it \Phi}_{\rm V}(\tau),$$
-*the&nbsp; [[Mobile_Communications/The_GWSSUS_Channel_Model#ACF_and_PDS_of_the_frequency-variant_transfer_function|$\text{frequency&ndash;Doppler's cross-power-spectral density}$]]:
+*the&nbsp; [[Mobile_Communications/The_GWSSUS_Channel_Model#ACF_and_PSD_of_the_frequency-variant_transfer_function|$\text{frequency&ndash;Doppler's cross-power-spectral density}$]]:
 :$${\it \Phi}_{\rm FD}(\Delta f, f_{\rm D})\hspace{0.3cm}\Rightarrow \hspace{0.3cm}\text{with}  \hspace{0.2cm}\Delta f = 0\text{:}  \hspace{0.2cm} {\it \Phi}_{\rm D}( f_{\rm D}),$$
-*the&nbsp; [[Mobile_Communications/The_GWSSUS_Channel_Model#ACF_and_PDS_of_the_delay-Doppler_function |$\text{delay&ndash;Doppler's cross-power-spectral density}$ ]]:
+*the&nbsp; [[Mobile_Communications/The_GWSSUS_Channel_Model#ACF_and_PSD_of_the_delay-Doppler_function |$\text{delay&ndash;Doppler's cross-power-spectral density}$ ]]:
 :$${\it \Phi}_{\rm VD}(\tau, f_{\rm D})= {\it \Phi}_{\rm V}(\tau) \cdot {\it \Phi}_{\rm D}(f_{\rm D})\hspace{0.05cm}.$$
 <br clear=all>