Aufgaben:Exercise 2.6: Dimensions in GWSSUS - Revision history

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

2022-02-17T11:41:20Z

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

Bene: Text replacement - "autocorrelation" to "auto-correlation"

2022-02-17T11:29:04Z

Text replacement - "autocorrelation" to "auto-correlation"

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

2022-02-17T11:20:08Z

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

Javier: Text replacement - "Category:Exercises for Mobile Communications" to "Category:Mobile Communications: Exercises"

2021-03-23T12:37:21Z

Text replacement - "Category:Exercises for Mobile Communications" to "Category:Mobile Communications: Exercises"

Guenter at 13:32, 14 January 2021

2021-01-14T13:32:38Z

Guenter at 13:06, 14 January 2021

2021-01-14T13:06:19Z

Guenter at 13:05, 14 January 2021

2021-01-14T13:05:16Z

Guenter at 13:03, 14 January 2021

2021-01-14T13:03:22Z

Javier: Text replacement - "===Sample solution===" to "===Solution==="

2020-12-16T13:16:40Z

Text replacement - "===Sample solution===" to "===Solution==="

Javier: Text replacement - "Mobile Kommunikation/Das GWSSUS–Kanalmodell" to "Mobile_Communications/The_GWSSUS_Channel_Model"

2020-12-15T14:01:07Z

Text replacement - "Mobile Kommunikation/Das GWSSUS–Kanalmodell" to "Mobile_Communications/The_GWSSUS_Channel_Model"

@@ Line 26: / Line 26: @@
   \hspace{0.2cm}  \stackrel{f, \hspace{0.02cm}\tau}{\bullet\!\!-\!\!\!-\!\!\!-\!\!\circ} \hspace{0.2cm} \eta_{\rm VZ}(\tau, t)\hspace{0.05cm}.$$
-*The correlation function&nbsp; $\varphi_{i_1\hspace{0.02cm}i_2}$&nbsp; and the power spectral density&nbsp; $\it \Phi_{i_1\hspace{0.02cm}i_2}$&nbsp; are provided with the same indices as the system function $\eta_{i_1\hspace{0.02cm}i_2}$.
+*The correlation function&nbsp; $\varphi_{i_1\hspace{0.02cm}i_2}$&nbsp; and the power-spectral density&nbsp; $\it \Phi_{i_1\hspace{0.02cm}i_2}$&nbsp; are provided with the same indices as the system function $\eta_{i_1\hspace{0.02cm}i_2}$.
 *Correlation functions can be recognized by the red font in the lower graph and all power densitiy spectra  are labeled in blue. The GWSSUS model is always assumed.
@@ Line 92: / Line 92: @@
 *Since the time-variant impulse response&nbsp; $\eta_{\rm VZ}(\tau, t)$&nbsp; has the unit&nbsp; $[1/\rm s]$, its ACF&nbsp; $\varphi_{\rm VZ}(\Delta \tau, \Delta t)$&nbsp; has the unit&nbsp; $[1/\rm s^2]$, both in the general case&nbsp; $\varphi_{\rm VZ}(\tau_1, l_1, \tau_2, t_2)$&nbsp; and with the GWSSUS case&nbsp; $\varphi_{\rm VZ}(\Delta \tau, \ \Delta t)$.
-*The Dirac function&nbsp; ${\rm \delta}(\Delta \tau)$&nbsp; has the unit&nbsp; $[1/\rm s]$, since the integral over all&nbsp; $\tau$&nbsp; $($with unit&nbsp; $[\rm s])$ must be&nbsp; $1$.&nbsp; Therefore, both the delay&ndash;time cross power spectral density&nbsp; ${\it \Phi}_{\rm VZ}(\tau, \Delta \tau)$&nbsp; and the delay power spectral density&nbsp; ${\it \Phi}_{\rm V}(\tau) = {\it \Phi}_{\rm VZ}(\tau, \Delta t = 0)$&nbsp; have unit&nbsp; $[1/\rm s]$.
+*The Dirac function&nbsp; ${\rm \delta}(\Delta \tau)$&nbsp; has the unit&nbsp; $[1/\rm s]$, since the integral over all&nbsp; $\tau$&nbsp; $($with unit&nbsp; $[\rm s])$ must be&nbsp; $1$.&nbsp; Therefore, both the delay&ndash;time cross power-spectral density&nbsp; ${\it \Phi}_{\rm VZ}(\tau, \Delta \tau)$&nbsp; and the delay power-spectral density&nbsp; ${\it \Phi}_{\rm V}(\tau) = {\it \Phi}_{\rm VZ}(\tau, \Delta t = 0)$&nbsp; have unit&nbsp; $[1/\rm s]$.
@@ Line 99: / Line 99: @@
 *The function&nbsp; ${\it \Phi}_{\rm VZ}(\tau, \Delta t)$&nbsp; has unit&nbsp; $[1/\rm s]$.&nbsp; Its Fourier transform with respect to&nbsp; $\tau$&nbsp; is&nbsp; $\varphi_{\rm FZ}(\Delta f, \Delta t)$, while its Fourier transform with respect to&nbsp; $t$&nbsp; is&nbsp; ${\it \Phi}_{\rm VD}(\tau, f_{\rm D})$.&nbsp; Both&nbsp; $\varphi_{\rm FZ}(\Delta f, \Delta t)$&nbsp; and&nbsp; ${\it \Phi}_{\rm VD}(\tau, f_{\rm D})$&nbsp; are therefore without unit.
-*The frequency&ndash;Doppler cross power spectral density has the unit&nbsp; $[\rm s] = [1/\rm Hz]$, because
+*The frequency&ndash;Doppler cross power-spectral density has the unit&nbsp; $[\rm s] = [1/\rm Hz]$, because
 :$${\it \Phi}_{\rm FD}(\Delta f, f_{\rm D}) = \int_{-\infty}^{+\infty} {\it \Phi}_{\rm VD}(\tau, f_{\rm D}) \cdot {\rm e}^{- {\rm j}\hspace{0.05cm}\cdot \hspace{0.05cm}2 \pi f_{\rm D} \tau}\hspace{0.15cm}{\rm d}\tau \hspace{0.05cm}. $$
 {{ML-Fuß}}

@@ Line 86: / Line 86: @@
 '''(2)'''&nbsp; <u>Solutions 2 and 3</u> are correct:
-*The autocorrelation&nbsp; $\rm (ACF)$&nbsp; function is by definition the following expected value:
+*The auto-correlation&nbsp; $\rm (ACF)$&nbsp; function is by definition the following expected value:
 :$$\varphi_{\rm VZ}(\tau_1, t_1, \tau_2, t_2) = {\rm E} \left [ \eta_{\rm VZ}(\tau_1, t_1) \cdot
   \eta_{\rm VZ}^{\star}(\tau_2, t_2) \right ]\hspace{0.05cm}.$$

@@ Line 26: / Line 26: @@
   \hspace{0.2cm}  \stackrel{f, \hspace{0.02cm}\tau}{\bullet\!\!-\!\!\!-\!\!\!-\!\!\circ} \hspace{0.2cm} \eta_{\rm VZ}(\tau, t)\hspace{0.05cm}.$$
-*The correlation function&nbsp; $\varphi_{i_1\hspace{0.02cm}i_2}$&nbsp; and the power density spectrum&nbsp; $\it \Phi_{i_1\hspace{0.02cm}i_2}$&nbsp; are provided with the same indices as the system function $\eta_{i_1\hspace{0.02cm}i_2}$.
+*The correlation function&nbsp; $\varphi_{i_1\hspace{0.02cm}i_2}$&nbsp; and the power spectral density&nbsp; $\it \Phi_{i_1\hspace{0.02cm}i_2}$&nbsp; are provided with the same indices as the system function $\eta_{i_1\hspace{0.02cm}i_2}$.
 *Correlation functions can be recognized by the red font in the lower graph and all power densitiy spectra  are labeled in blue. The GWSSUS model is always assumed.
@@ Line 92: / Line 92: @@
 *Since the time-variant impulse response&nbsp; $\eta_{\rm VZ}(\tau, t)$&nbsp; has the unit&nbsp; $[1/\rm s]$, its ACF&nbsp; $\varphi_{\rm VZ}(\Delta \tau, \Delta t)$&nbsp; has the unit&nbsp; $[1/\rm s^2]$, both in the general case&nbsp; $\varphi_{\rm VZ}(\tau_1, l_1, \tau_2, t_2)$&nbsp; and with the GWSSUS case&nbsp; $\varphi_{\rm VZ}(\Delta \tau, \ \Delta t)$.
-*The Dirac function&nbsp; ${\rm \delta}(\Delta \tau)$&nbsp; has the unit&nbsp; $[1/\rm s]$, since the integral over all&nbsp; $\tau$&nbsp; $($with unit&nbsp; $[\rm s])$ must be&nbsp; $1$.&nbsp; Therefore, both the delay&ndash;time cross power density spectrum&nbsp; ${\it \Phi}_{\rm VZ}(\tau, \Delta \tau)$&nbsp; and the delay power density spectrum&nbsp; ${\it \Phi}_{\rm V}(\tau) = {\it \Phi}_{\rm VZ}(\tau, \Delta t = 0)$&nbsp; have unit&nbsp; $[1/\rm s]$.
+*The Dirac function&nbsp; ${\rm \delta}(\Delta \tau)$&nbsp; has the unit&nbsp; $[1/\rm s]$, since the integral over all&nbsp; $\tau$&nbsp; $($with unit&nbsp; $[\rm s])$ must be&nbsp; $1$.&nbsp; Therefore, both the delay&ndash;time cross power spectral density&nbsp; ${\it \Phi}_{\rm VZ}(\tau, \Delta \tau)$&nbsp; and the delay power spectral density&nbsp; ${\it \Phi}_{\rm V}(\tau) = {\it \Phi}_{\rm VZ}(\tau, \Delta t = 0)$&nbsp; have unit&nbsp; $[1/\rm s]$.
@@ Line 99: / Line 99: @@
 *The function&nbsp; ${\it \Phi}_{\rm VZ}(\tau, \Delta t)$&nbsp; has unit&nbsp; $[1/\rm s]$.&nbsp; Its Fourier transform with respect to&nbsp; $\tau$&nbsp; is&nbsp; $\varphi_{\rm FZ}(\Delta f, \Delta t)$, while its Fourier transform with respect to&nbsp; $t$&nbsp; is&nbsp; ${\it \Phi}_{\rm VD}(\tau, f_{\rm D})$.&nbsp; Both&nbsp; $\varphi_{\rm FZ}(\Delta f, \Delta t)$&nbsp; and&nbsp; ${\it \Phi}_{\rm VD}(\tau, f_{\rm D})$&nbsp; are therefore without unit.
-*The frequency&ndash;Doppler cross power density spectrum has the unit&nbsp; $[\rm s] = [1/\rm Hz]$, because
+*The frequency&ndash;Doppler cross power spectral density has the unit&nbsp; $[\rm s] = [1/\rm Hz]$, because
 :$${\it \Phi}_{\rm FD}(\Delta f, f_{\rm D}) = \int_{-\infty}^{+\infty} {\it \Phi}_{\rm VD}(\tau, f_{\rm D}) \cdot {\rm e}^{- {\rm j}\hspace{0.05cm}\cdot \hspace{0.05cm}2 \pi f_{\rm D} \tau}\hspace{0.15cm}{\rm d}\tau \hspace{0.05cm}. $$
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← Older revision		Revision as of 12:37, 23 March 2021
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−	[[Category:~~Exercises for~~ Mobile Communications\|^2.3 The GWSSUS Channel Model^]]	+	[[Category:Mobile Communications: Exercises\|^2.3 The GWSSUS Channel Model^]]

@@ Line 49: / Line 49: @@
 {Which of the following specified dimensions of the system functions are correct?
 |type="[]"}
-+ $\eta_{\rm VZ}(\tau, t)$&nbsp; has the dimension&nbsp; $[1/\rm s]$.
++ $\eta_{\rm VZ}(\tau, t)$&nbsp; has the unit&nbsp; $[1/\rm s]$.
-+ $\eta_{\rm FZ}(f, t)$&nbsp; is dimensionless.
++ $\eta_{\rm FZ}(f, t)$&nbsp; is without unit.
-+ $\eta_{\rm VD}(\tau, f_{\rm D})$&nbsp; is dimensionless.
++ $\eta_{\rm VD}(\tau, f_{\rm D})$&nbsp; is without unit.
-+ $\eta_{\rm FD}(f, f_{\rm D})$&nbsp; has the dimension&nbsp; $[1/\rm Hz]$.
++ $\eta_{\rm FD}(f, f_{\rm D})$&nbsp; has the unit&nbsp; $[1/\rm Hz]$.
 {Which of the following statements are correct?
 |type="[]"}
-- $\varphi_{\rm VZ}(\Delta \tau, \Delta t)$&nbsp; has the dimension&nbsp; $[1/\rm s]$.
+- $\varphi_{\rm VZ}(\Delta \tau, \Delta t)$&nbsp; has the unit&nbsp; $[1/\rm s]$.
-+ ${\it \phi}_{\rm VZ}(\tau, {\rm \Delta} t)$&nbsp; has the dimension&nbsp; $[1/\rm s]$.
++ ${\it \phi}_{\rm VZ}(\tau, {\rm \Delta} t)$&nbsp; has the unit&nbsp; $[1/\rm s]$.
-+ ${\it \phi}_{\rm V}(\tau)$&nbsp; has the dimension&nbsp; $[1/\rm s]$.
++ ${\it \phi}_{\rm V}(\tau)$&nbsp; has the unit&nbsp; $[1/\rm s]$.
 {Which of the following statements  are correct?
 |type="[]"}
-+ $\varphi_{\rm FZ}(\Delta f, \Delta t),&nbsp; \varphi_{\rm F}(\Delta f)$&nbsp; and&nbsp; $\varphi_{\rm Z}(\Delta t)$&nbsp; have no dimension.
++ $\varphi_{\rm FZ}(\Delta f, \Delta t),&nbsp; \varphi_{\rm F}(\Delta f)$&nbsp; and&nbsp; $\varphi_{\rm Z}(\Delta t)$&nbsp; have no unit.
-- ${\it \Phi}_{\rm VD}(\tau, f_{\rm D})$&nbsp; has the dimension&nbsp; $[1/\rm s]$.
+- ${\it \Phi}_{\rm VD}(\tau, f_{\rm D})$&nbsp; has the unit&nbsp; $[1/\rm s]$.
-+ ${\it \Phi}_{\rm FD}(\Delta f, f_{\rm D})$&nbsp; and&nbsp; ${\it \Phi}_{\rm D}(f_{\rm D})$&nbsp; have the dimension $[1/\rm Hz]$ each.
++ ${\it \Phi}_{\rm FD}(\Delta f, f_{\rm D})$&nbsp; and&nbsp; ${\it \Phi}_{\rm D}(f_{\rm D})$&nbsp; have the unit&nbsp; $[1/\rm Hz]$ each.
 </quiz>
@@ Line 70: / Line 70: @@
 {{ML-Kopf}}
 '''(1)'''&nbsp; <u>All statements are correct</u>:
-*$\eta_{\rm VZ}(\tau, t)$ is the time-variant impulse response, for which the term $h(\tau, t)$ is also common. Like every impulse response, $h(\tau, t)$ has the unit $[1/\rm s]$.
+*$\eta_{\rm VZ}(\tau, t)$&nbsp; is the time-variant impulse response, for which the term&nbsp; $h(\tau, t)$&nbsp; is also common.&nbsp; Like every impulse response,&nbsp; $h(\tau, t)$&nbsp; has the unit&nbsp; $[1/\rm s]$.
-*By Fourier transformation of the function $\eta_{\rm VZ}(\tau, t)$ with respect to the delay $\tau$ one obtains
+*By Fourier transform of the function&nbsp; $\eta_{\rm VZ}(\tau, t)$&nbsp; with respect to the delay&nbsp; $\tau$&nbsp; one obtains
 :$$\eta_{\rm FZ}(f, t) = \int_{-\infty}^{+\infty} \eta_{\rm VZ}(\tau, t) \cdot {\rm e}^{- {\rm j}\hspace{0.05cm}\cdot \hspace{0.05cm}2 \pi f \tau}\hspace{0.15cm}{\rm d}\tau
   \hspace{0.05cm}. $$
-*Due to the integration over $\tau$ (unit: $\rm s$), the time-variant transfer function $\eta_{\rm FZ}(f, t)$ is dimensionless. In some literature, $H(f, t)$ is also used instead of $\eta_{\rm FZ}(f, t)$.
+*Due to the integration over&nbsp; $\tau$&nbsp; $($unit:&nbsp; $\rm s)$, the time-variant transfer function&nbsp; $\eta_{\rm FZ}(f, t)$&nbsp; is dimensionless.&nbsp; <br>In some literature,&nbsp; $H(f, t)$&nbsp; is also used instead of&nbsp; $\eta_{\rm FZ}(f, t)$.
-*The delay&ndash;Doppler representation $\eta_{\rm VD}(\tau, f_{\rm D})$ is also dimensionless. This function results from the time-variant impulse response $\eta_{\rm VZ}(\tau, t)$ by Fourier transformation with respect to $t$:
+*The delay&ndash;Doppler representation&nbsp; $\eta_{\rm VD}(\tau, f_{\rm D})$&nbsp; is dimensionless, too.&nbsp; This function results from the time-variant impulse response&nbsp; $\eta_{\rm VZ}(\tau, t)$&nbsp; by Fourier transform with respect to&nbsp; $t$:
 :$$\eta_{\rm VD}(\tau, f_{\rm D}) = \int_{-\infty}^{+\infty} \eta_{\rm VZ}(\tau, t) \cdot {\rm e}^{- {\rm j}\hspace{0.05cm}\cdot \hspace{0.05cm}2 \pi f_{\rm D} t}\hspace{0.15cm}{\rm d}t
   \hspace{0.05cm}.$$
-*The function $\eta_{\rm FD}(t, f_{\rm D})$ is obtained from the dimensionless functions $\eta_{\rm VD}(\tau, f_{\rm D})$ and $\eta_{\rm FZ}(f, t)$ respectively by a Fourier transformation, which results in the unit $[\rm s] = [1/\rm Hz]$.
+*The function&nbsp; $\eta_{\rm FD}(t, f_{\rm D})$&nbsp; is obtained from the dimensionless functions&nbsp; $\eta_{\rm VD}(\tau, f_{\rm D})$&nbsp; and&nbsp; $\eta_{\rm FZ}(f, t)$&nbsp; respectively by a Fourier transform, which results in the unit&nbsp; $[\rm s] = [1/\rm Hz]$.
 '''(2)'''&nbsp; <u>Solutions 2 and 3</u> are correct:
-*The autocorrelation (ACF) function is by definition the following expected value:
+*The autocorrelation&nbsp; $\rm (ACF)$&nbsp; function is by definition the following expected value:
 :$$\varphi_{\rm VZ}(\tau_1, t_1, \tau_2, t_2) = {\rm E} \left [ \eta_{\rm VZ}(\tau_1, t_1) \cdot
   \eta_{\rm VZ}^{\star}(\tau_2, t_2) \right ]\hspace{0.05cm}.$$
-*Since the time-variant impulse response $\eta_{\rm VZ}(\tau, t)$ has the unit $[1/\rm s]$, its ACF $\varphi_{\rm VZ}$ has the unit $[1/\rm s^2]$, both in the general case $\varphi_{\rm VZ}(\tau_1, l_1, \tau_2, t_2)$ and with the GWSSUS case $\varphi_{\rm VZ}(\delta \tau, \ \delta t)$.
+*Since the time-variant impulse response&nbsp; $\eta_{\rm VZ}(\tau, t)$&nbsp; has the unit&nbsp; $[1/\rm s]$, its ACF&nbsp; $\varphi_{\rm VZ}(\Delta \tau, \Delta t)$&nbsp; has the unit&nbsp; $[1/\rm s^2]$, both in the general case&nbsp; $\varphi_{\rm VZ}(\tau_1, l_1, \tau_2, t_2)$&nbsp; and with the GWSSUS case&nbsp; $\varphi_{\rm VZ}(\Delta \tau, \ \Delta t)$.
-*The Dirac function $\delta(\Delta \tau)$ has the dimension $[1/\rm s]$, since the integral over all $\tau$ (with dimension $[\rm s]$) must be $1$. Therefore, both the delay&ndash;time cross power density spectrum ${\it \Phi}_{\rm VZ}(\tau, \delta \tau)$ and the delay power density spectrum ${\it \Phi}_{\rm V}(\tau) = {\it \Phi}_{\rm VZ}(\tau, \delta t = 0)$ have dimension $[1/\rm s]$.
+*The Dirac function&nbsp; ${\rm \delta}(\Delta \tau)$&nbsp; has the unit&nbsp; $[1/\rm s]$, since the integral over all&nbsp; $\tau$&nbsp; $($with unit&nbsp; $[\rm s])$ must be&nbsp; $1$.&nbsp; Therefore, both the delay&ndash;time cross power density spectrum&nbsp; ${\it \Phi}_{\rm VZ}(\tau, \Delta \tau)$&nbsp; and the delay power density spectrum&nbsp; ${\it \Phi}_{\rm V}(\tau) = {\it \Phi}_{\rm VZ}(\tau, \Delta t = 0)$&nbsp; have unit&nbsp; $[1/\rm s]$.
 '''(3)'''&nbsp; <u>Statements 1 and 3</u> are correct:
-*The function ${\it \Phi}_{\rm VZ}(\tau, \delta t)$ has dimension $[1/\rm s]$. Its Fourier transform with respect to $\tau$ is $\varphi_{\rm FZ}(\Delta f, \Delta t)$, while its Fourier transform with respect to $t$ is ${\it \Phi}_{\rm VD}(\tau, f_{\rm D})$. Both $\varphi_{\rm FZ}(\Delta f, \Delta t)$ and ${\it \Phi}_{\rm VD}(\tau, f_{\rm D})$ are therefore dimensionless.
+*The function&nbsp; ${\it \Phi}_{\rm VZ}(\tau, \Delta t)$&nbsp; has unit&nbsp; $[1/\rm s]$.&nbsp; Its Fourier transform with respect to&nbsp; $\tau$&nbsp; is&nbsp; $\varphi_{\rm FZ}(\Delta f, \Delta t)$, while its Fourier transform with respect to&nbsp; $t$&nbsp; is&nbsp; ${\it \Phi}_{\rm VD}(\tau, f_{\rm D})$.&nbsp; Both&nbsp; $\varphi_{\rm FZ}(\Delta f, \Delta t)$&nbsp; and&nbsp; ${\it \Phi}_{\rm VD}(\tau, f_{\rm D})$&nbsp; are therefore without unit.
-*The frequency&ndash;Doppler cross power density spectrum has the unit $[\rm s] = [1/\rm Hz]$, because
+*The frequency&ndash;Doppler cross power density spectrum has the unit&nbsp; $[\rm s] = [1/\rm Hz]$, because
-:$${\it \Phi}_{\rm FD}(\delta f, f_{\rm D}) = \int_{-\infty}^{+\infty} {\it \Phi}_{\rm VD}(\tau, f_{\rm D}) \cdot {\rm e}^{- {\rm j}\hspace{0.05cm}\cdot \hspace{0.05cm}2 \pi f_{\rm D} \tau}\hspace{0.15cm}{\rm d}\tau \hspace{0.05cm}. $$
+:$${\it \Phi}_{\rm FD}(\Delta f, f_{\rm D}) = \int_{-\infty}^{+\infty} {\it \Phi}_{\rm VD}(\tau, f_{\rm D}) \cdot {\rm e}^{- {\rm j}\hspace{0.05cm}\cdot \hspace{0.05cm}2 \pi f_{\rm D} \tau}\hspace{0.15cm}{\rm d}\tau \hspace{0.05cm}. $$
 {{ML-Fuß}}

@@ Line 17: / Line 17: @@
 The relationship between the functions is shown in the diagram (yellow background).&nbsp; The Fourier correspondences are shown in green:
 * The transition from a circle filled with white to a circle filled with green corresponds to the&nbsp; [[Signal_Representation/Fourier_Transform_and_Its_Inverse#The_First_Fourier_Integral|Fourier transform]].
-* The transition from a circle filled with green to a circle filled with white corresponds to the&nbsp; [[Signal_Representation/Fourier_Transform_and_Its_Inverse#The_Second_Fourier_Integral]]&nbsp; (opposite direction).
+* The transition from a circle filled with green to a circle filled with white corresponds to the&nbsp; [[Signal_Representation/Fourier_Transform_and_Its_Inverse#The_Second_Fourier_Integral|inverse Fourier transform]]&nbsp; (opposite direction).

@@ Line 9: / Line 9: @@
 We denote all of the functions with&nbsp; $\eta_{i_1i_2}$. The indices $i_1$ and $i_2$ are defined as follows:
-* <b>V</b>&nbsp; stands for delay&nbsp; $\tau$&nbsp; (index $i_1$),
+* $\boldsymbol{\rm V}$&nbsp; $($because of German&nbsp; $\rm V\hspace{-0.05cm}$erzögerung$)$&nbsp; stands for delay time&nbsp; $\tau$&nbsp; $($index&nbsp; $i_1)$,
-* <b>F</b>&nbsp; stands for frequency&nbsp; $f$&nbsp; (index $i_1$),
+* $\boldsymbol{\rm F}$&nbsp;  stands for frequency&nbsp; $f$&nbsp; $($index&nbsp; $i_1)$,
-* <b>Z</b>&nbsp; stands for the time&nbsp; $t$&nbsp; (index $i_2$)
+* $\boldsymbol{\rm Z}$&nbsp; $($because of German&nbsp; $\rm Z\hspace{-0.05cm}$eit$)$&nbsp; stands for the time&nbsp; $t$&nbsp; $($index&nbsp; $i_2)$,
-* <b>D</b>&nbsp; stands for the Doppler frequency&nbsp; $f_{\rm D}$&nbsp; (index $i_2$).
+* $\boldsymbol{\rm D}$&nbsp; stands for the Doppler frequency $f_{\rm D}$&nbsp; $($index&nbsp; $i_2)$.
-The relationship between the functions is shown in the diagram (yellow background). The Fourier correspondences are shown in green:
+The relationship between the functions is shown in the diagram (yellow background).&nbsp; The Fourier correspondences are shown in green:
-* The transition from a circle filled with white to a circle filled with green corresponds to a&nbsp; [[Signal_Representation/Fourier_Transform_and_Its_Inverse#Das_erste_Fourierintegral|Fourier transform]].
+* The transition from a circle filled with white to a circle filled with green corresponds to the&nbsp; [[Signal_Representation/Fourier_Transform_and_Its_Inverse#Das_erste_Fourierintegral|Fourier transform]].
 * The transition from a circle filled with green to a circle filled with white corresponds to the&nbsp; [[Signal_Representation/Fourier_Transform_and_Its_Inverse#Das_zweite_Fourierintegral|inverse Fourier transform]]&nbsp; (opposite direction).
@@ Line 26: / Line 26: @@
   \hspace{0.2cm}  \stackrel{f, \hspace{0.02cm}\tau}{\bullet\!\!-\!\!\!-\!\!\!-\!\!\circ} \hspace{0.2cm} \eta_{\rm VZ}(\tau, t)\hspace{0.05cm}.$$
-*The correlation function&nbsp; $\varphi_{i_1i_2}$&nbsp; and the power density spectrum&nbsp; $\it \Phi_{i_1i_2}$&nbsp; are provided with the same indices as the system function $\eta_{i_1i_2}$.
+*The correlation function&nbsp; $\varphi_{i_1\hspace{0.02cm}i_2}$&nbsp; and the power density spectrum&nbsp; $\it \Phi_{i_1\hspace{0.02cm}i_2}$&nbsp; are provided with the same indices as the system function $\eta_{i_1\hspace{0.02cm}i_2}$.
-*Correlation functions can be recognized by the red font in the lower graph and all power spectral densities are labeled in blue. The GWSSUS model is always assumed.
+*Correlation functions can be recognized by the red font in the lower graph and all power densitiy spectra  are labeled in blue. The GWSSUS model is always assumed.
-Let us consider here the system function&nbsp; $\eta_{\rm VZ}(\tau, t)$, i.e. the time variant impulse response&nbsp; $h(\tau, t)$. We define:
+Let us consider here the system function&nbsp; $\eta_{\rm VZ}(\tau, t)$, i.e. the time&ndash;variant impulse response&nbsp; $h(\tau, t)$.&nbsp; We define:
 :$$\varphi_{\rm VZ}(\tau_1, t_1, \tau_2, t_2) = {\rm E} \left [ \eta_{\rm VZ}(\tau_1, t_1) \cdot
   \eta_{\rm VZ}^{\star}(\tau_2, t_2) \right ]\hspace{0.05cm},$$
@@ Line 41: / Line 41: @@
-''Note:'' &nbsp; This task belongs to the chapter&nbsp; [[Mobile_Communications/Das_GWSSUS%E2%80%93Kanalmodell| Das GWSUS&ndash;Kanalmodell]].
+''Note:'' &nbsp; This exercise belongs to the chapter&nbsp; [[Mobile_Communications/The_GWSSUS_Channel_Model| The GWSSUS Channel Model]].
@@ Line 49: / Line 49: @@
 {Which of the following specified dimensions of the system functions are correct?
 |type="[]"}
-+ $\eta_{\rm VZ}(\tau, t)$&nbsp; has the dimension&nbsp; $[1/\rm s]$
++ $\eta_{\rm VZ}(\tau, t)$&nbsp; has the dimension&nbsp; $[1/\rm s]$.
 + $\eta_{\rm FZ}(f, t)$&nbsp; is dimensionless.
 + $\eta_{\rm VD}(\tau, f_{\rm D})$&nbsp; is dimensionless.
-+ $\eta_{\rm FD}(f, f_{\rm D})$&nbsp; has the dimension&nbsp; $[1/\rm Hz]$
++ $\eta_{\rm FD}(f, f_{\rm D})$&nbsp; has the dimension&nbsp; $[1/\rm Hz]$.
-{Which of the following are correct?
+{Which of the following statements are correct?
 |type="[]"}
-- $\varphi_{\rm VZ}(\delta \tau, \delta t)$&nbsp; has the dimension&nbsp; $[1/\rm s]$.
+- $\varphi_{\rm VZ}(\Delta \tau, \Delta t)$&nbsp; has the dimension&nbsp; $[1/\rm s]$.
-+ ${\it \phi}_{\rm VZ}(\tau, {\rm \delta} t)$&nbsp; has the dimension&nbsp; $[1/\rm s]$.
++ ${\it \phi}_{\rm VZ}(\tau, {\rm \Delta} t)$&nbsp; has the dimension&nbsp; $[1/\rm s]$.
 + ${\it \phi}_{\rm V}(\tau)$&nbsp; has the dimension&nbsp; $[1/\rm s]$.
-{Which of the following are correct?
+{Which of the following statements  are correct?
 |type="[]"}
 + $\varphi_{\rm FZ}(\Delta f, \Delta t),&nbsp; \varphi_{\rm F}(\Delta f)$&nbsp; and&nbsp; $\varphi_{\rm Z}(\Delta t)$&nbsp; have no dimension.

@@ Line 67: / Line 67: @@
 </quiz>
-===Sample solution===
+===Solution===
 {{ML-Kopf}}
 '''(1)'''&nbsp; <u>All statements are correct</u>:

← Older revision		Revision as of 14:01, 15 December 2020
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−	{{quiz-Header\|Buchseite=~~Mobile Kommunikation~~/~~Das GWSSUS–Kanalmodell~~}}	+	{{quiz-Header\|Buchseite=Mobile_Communications/The_GWSSUS_Channel_Model}}

	[[File:P_ID2167__Mob_A_2_6.png\|right\|frame\|Overview of the GWSSUS functions]]		[[File:P_ID2167__Mob_A_2_6.png\|right\|frame\|Overview of the GWSSUS functions]]